import { type PracticeQuestion } from "../../types/lesson";

export const EQUIV_EXPR_EASY: PracticeQuestion[] = [
  {
    id: "0354c7de",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the given expression?",
    choices: [
      { label: "A", text: "<strong>5 · (x + 3, )</strong>" },
      { label: "B", text: "<strong>5 · (x + 10, )</strong>" },
      { label: "C", text: "<strong>5 · (x + 15, )</strong>" },
      { label: "D", text: "<strong>5 · (x + 20, )</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since 5 is a factor of both terms, <strong>5 x</strong> and 15, the given expression can be factored and rewritten as <strong>5 · (x + 3, )</strong>.Choice B is incorrect and may result from subtracting 5 from the constant when factoring 5 from the given expression. Choice C is incorrect and may result from factoring 5 from only the first term, not both terms, of the given expression. Choice D is incorrect and may result from adding 5 to the constant when factoring 5 from the given expression.",
    hasFigure: false,
  },
  {
    id: "0536ad4f",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>20 w − (4 w + 3 w)</strong>?",
    choices: [
      { label: "A", text: "<strong>10 w</strong>" },
      { label: "B", text: "<strong>13 w</strong>" },
      { label: "C", text: "<strong>19 w</strong>" },
      { label: "D", text: "<strong>21 w</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Combining like terms inside the parentheses of the given expression, <strong>20 w − (4 w + 3 w)</strong>, yields <strong>20 w − (7 w)</strong>. Combining like terms in this resulting expression yields <strong>13 w</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "127b2759",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>8 + d² + 3</strong>?",
    choices: [
      { label: "A", text: "<strong>d² + 24</strong>" },
      { label: "B", text: "<strong>d² + 11</strong>" },
      { label: "C", text: "<strong>d² + 5</strong>" },
      { label: "D", text: "<strong>d² − 11</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The given expression can be rewritten as <strong>d² + 8 + 3</strong>. Adding <strong>8</strong> and <strong>3</strong> in this expression yields <strong>d² + 11</strong>.<br>Choice A is incorrect. This expression is equivalent to <strong>d² + 8 (3)</strong>.<br>Choice C is incorrect. This expression is equivalent to <strong>8 + d² − 3</strong>.<br>Choice D is incorrect. This expression is equivalent to <strong>−8 + d² − 3</strong>.",
    hasFigure: false,
  },
  {
    id: "1d3fee25",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to <strong>3 · (x + 5, ) − 6</strong> ?",
    choices: [
      { label: "A", text: "<strong>3 x − 3</strong>" },
      { label: "B", text: "<strong>3 x − 1</strong>" },
      { label: "C", text: "<strong>3 x + 9</strong>" },
      { label: "D", text: "<strong>15 x − 6</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Using the distributive property to multiply 3 and <strong>(x + 5, )</strong> gives <strong>3 x + 15 − 6</strong>, which can be rewritten as <strong>3 x + 9</strong>.Choice A is incorrect and may result from rewriting the given expression as <strong>3 · (x + 5 − 6, )</strong>. Choice B is incorrect and may result from incorrectly rewriting the expression as <strong>(3 x + 5, ) − 6</strong>. Choice D is incorrect and may result from incorrectly rewriting the expression as <strong>3 · 5 x − 6</strong>.",
    hasFigure: false,
  },
  {
    id: "1e8d7183",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>256 w² − 676</strong>?",
    choices: [
      { label: "A", text: "<strong>(16 w − 26) (16 w − 26)</strong>" },
      { label: "B", text: "<strong>(8 w − 13) (8 w + 13)</strong>" },
      { label: "C", text: "<strong>(8 w − 13) (8 w − 13)</strong>" },
      { label: "D", text: "<strong>(16 w − 26) (16 w + 26)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The given expression follows the difference of two squares pattern, <strong>x² − y²</strong>, which factors as <strong>(x − y) (x + y)</strong>. Therefore, the expression <strong>256 w² − 676</strong> can be written as <strong>(16 w)² − 26²</strong>, or <strong>(16 w) (16 w) − (26) (26)</strong>, which factors as <strong>(16 w − 26) (16 w + 26)</strong>.<br>Choice A is incorrect. This expression is equivalent to <strong>256 w² − 832 w + 676</strong>.<br>Choice B is incorrect. This expression is equivalent to <strong>64 w² − 169</strong>.<br>Choice C is incorrect. This expression is equivalent to <strong>64 w² − 208 w + 169</strong>.",
    hasFigure: false,
  },
  {
    id: "294db8ec",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to <strong>2 x³ + 4</strong>?",
    choices: [
      { label: "A", text: "<strong>4 · (x³ + 4, )</strong>" },
      { label: "B", text: "<strong>4 · (x³ + 2, )</strong>" },
      { label: "C", text: "<strong>2 · (x³ + 4, )</strong>" },
      { label: "D", text: "<strong>2 · (x³ + 2, )</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The expression <strong>2 x³ + 4</strong> has two terms, <strong>2 x³</strong> and 4. The greatest common factor of these two terms is 2. Factoring 2 from each of these terms yields <strong>2 · x³ + 2 · 2</strong>, or <strong>2 · (x³ + 2, )</strong>.Choices A and B are incorrect because 4 is not a factor of the term <strong>2 x³</strong>. Choice C is incorrect and may result from factoring 2 from <strong>2 x³</strong> but not from 4.",
    hasFigure: false,
  },
  {
    id: "499cb491",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>5 x² − 50 xy²</strong>?",
    choices: [
      { label: "A", text: "<strong>5 x (x − 10 y²)</strong>" },
      { label: "B", text: "<strong>5 x (x − 50 y²)</strong>" },
      { label: "C", text: "<strong>5 x² (10 xy²)</strong>" },
      { label: "D", text: "<strong>5 x² (50 xy²)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since each term of the given expression has a factor of <strong>5 x</strong>, it can be rewritten as <strong>5 x (x) − 5 x (10 y²)</strong>, or <strong>5 x (x − 10 y²)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "49efde89",
    type: "mcq",
    questionHtml:
      "The expression <strong>2 x² + a, x</strong> is equivalent to <strong>x · (2 x + 7, )</strong> for some constant a. What is the value of a ?",
    choices: [
      { label: "A", text: "2" },
      { label: "B", text: "3" },
      { label: "C", text: "4" },
      { label: "D", text: "7" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that <strong>2 x² + a, x</strong> is equivalent to <strong>x · (2 x + 7, )</strong> for some constant a. Distributing the x over each term in the parentheses gives <strong>2 x² + 7 x</strong>, which is in the same form as the first given expression, <strong>2 x² + a, x</strong>. The coefficient of the second term in <strong>2 x² + 7 x</strong> is 7. Therefore, the value of a is 7.Choice A is incorrect. If the value of a were 2, then <strong>2 x² + a, x</strong> would be equivalent to <strong>2 x² + 2 x</strong>, which isn’t equivalent to <strong>x · (2 x + 7, )</strong>. Choice B is incorrect. If the value of a were 3, then <strong>2 x² + a, x</strong> would be equivalent to <strong>2 x² + 3 x</strong>, which isn’t equivalent to <strong>x · (2 x + 7, )</strong>. Choice C is incorrect. If the value of a were 4, then <strong>2 x² + a, x</strong> would be equivalent to <strong>2 x² + 4 x</strong>, which isn’t equivalent to <strong>x · (2 x + 7, )</strong>.",
    hasFigure: false,
  },
  {
    id: "4a5af623",
    type: "mcq",
    questionHtml:
      "Which expression is a factor of <strong>2 x² + 38 x + 10</strong>?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>5 x</strong>" },
      { label: "C", text: "<strong>38 x</strong>" },
      { label: "D", text: "<strong>2 x²</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since <strong>2</strong> is a common factor of each of the terms in the given expression, the expression can be rewritten as <strong>2 (x² + 19 x + 5)</strong>. Therefore, the factors of the given expression are <strong>2</strong> and <strong>x² + 19 x + 5</strong>. Of these two factors, only <strong>2</strong> is listed as a choice.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is a term of the given expression, not a factor of the given expression.<br>Choice D is incorrect. This is a term of the given expression, not a factor of the given expression.",
    hasFigure: false,
  },
  {
    id: "5d93c782",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>x² + 3 x − 40</strong>?",
    choices: [
      { label: "A", text: "<strong>(x − 4) (x + 10)</strong>" },
      { label: "B", text: "<strong>(x − 5) (x + 8)</strong>" },
      { label: "C", text: "<strong>(x − 8) (x + 5)</strong>" },
      { label: "D", text: "<strong>(x − 10) (x + 4)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The given expression may be rewritten as <strong>x² + 8 x − 5 x − 40</strong>. Since the first two terms of this expression have a common factor of <strong>x</strong> and the last two terms of this expression have a common factor of <strong>−5</strong>, this expression may be rewritten as <strong>x (x) + x (8) − 5 (x) − 5 (8)</strong>, or <strong>x (x + 8) − 5 (x + 8)</strong>. Since each term of this expression has a common factor of <strong>(x + 8)</strong>, it may be rewritten as <strong>(x − 5) (x + 8)</strong>.<br>Alternate approach: An expression of the form <strong>x² + b x + c</strong>, where <strong>b</strong> and <strong>c</strong> are constants, can be factored if there are two values that add to give <strong>b</strong> and multiply to give <strong>c</strong>. In the given expression, <strong>b = 3</strong> and <strong>c = −40</strong>. The values of <strong>−5</strong> and <strong>8</strong> add to give <strong>3</strong> and multiply to give <strong>−40</strong>, so the expression can be factored as <strong>(x − 5) (x + 8)</strong>.<br>Choice A is incorrect. This expression is equivalent to <strong>x² + 6 x − 40</strong>, not <strong>x² + 3 x − 40</strong>.<br>Choice C is incorrect. This expression is equivalent to <strong>x² − 3 x − 40</strong>, not <strong>x² + 3 x − 40</strong>.<br>Choice D is incorrect. This expression is equivalent to <strong>x² − 6 x − 40</strong>, not <strong>x² + 3 x − 40</strong>.",
    hasFigure: false,
  },
  {
    id: "60fdb4d4",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>parenthesis, 2 x² − 4, ) − parenthesis −3 x² + 2 x − 7, )</strong> ?",
    choices: [
      { label: "A", text: "<strong>5 x² − 2 x + 3</strong>" },
      { label: "B", text: "<strong>5 x² + 2 x − 3</strong>" },
      { label: "C", text: "<strong>−x² − 2 x − 11</strong>" },
      { label: "D", text: "<strong>−x² + 2 x − 11</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The given expression <strong>(2 x² − 4, ) − (−3, x² + 2 x − 7, )</strong> can be rewritten as <strong>2 x² − 4 + 3 x² − 2 x + 7</strong>. Combining like terms yields <strong>5 x² − 2 x + 3</strong>.Choices B, C, and D are incorrect and may be the result of errors when applying the distributive property.",
    hasFigure: false,
  },
  {
    id: "67e866b5",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>9 x² + 7 x² + 9 x</strong>?",
    choices: [
      { label: "A", text: "<strong>63 x^(4 + 9 x)</strong>" },
      { label: "B", text: "<strong>9 x² + 16 x</strong>" },
      { label: "C", text: "<strong>25 x⁵</strong>" },
      { label: "D", text: "<strong>16 x² + 9 x</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. In the given expression, the first two terms, <strong>9 x²</strong> and <strong>7 x²</strong>, are like terms. Combining these like terms yields <strong>9 x² + 7 x²</strong>, or <strong>16 x²</strong>. It follows that the expression <strong>9 x² + 7 x² + 9 x</strong> is equivalent to <strong>16 x² + 9 x</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "6e06a0a7",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to <strong>2, a² · (a + 3, )</strong> ?",
    choices: [
      { label: "A", text: "<strong>5, a³</strong>" },
      { label: "B", text: "<strong>8, a to the fifth power</strong>" },
      { label: "C", text: "<strong>2, a³ + 3</strong>" },
      { label: "D", text: "<strong>2, a³ + 6, a²</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Expanding the given expression using the distributive property yields <strong>2 a, ² · a + 2 a, ² · 3</strong>. Combining like terms yields <strong>2 a, ² · (a, to the first power, ) + (2 · 3, ) · (a, ², )</strong>, or <strong>2 a, raised to the 2 + 1 power + 6 a, ²</strong>, which is equivalent to <strong>2 a, ³ + 6 a, ²</strong>.Choices A and B are incorrect and may result from incorrectly combining like terms. Choice C is incorrect and may result from distributing <strong>2 a, ²</strong> only to a, and not to 3, in the given expression.",
    hasFigure: false,
  },
  {
    id: "70482e20",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>11 x³ − 5 x³</strong>?",
    choices: [
      { label: "A", text: "<strong>16 x³</strong>" },
      { label: "B", text: "<strong>6 x³</strong>" },
      { label: "C", text: "<strong>6 x⁶</strong>" },
      { label: "D", text: "<strong>16 x⁶</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The given expression can be rewritten as <strong>11 x³ + (−5) x³</strong>. Since the two terms of this expression are both constant multiples of <strong>x³</strong>, they are like terms and can, therefore, be combined through addition. Adding like terms in the expression <strong>11 x³ + (−5) x³</strong> yields <strong>6 x³</strong>.<br>Choice A is incorrect. This is equivalent to <strong>11 x³ + 5 x³</strong>, not <strong>11 x³ − 5 x³</strong>.<br>Choice C is incorrect. This is equivalent to <strong>11 x^(6 − 5 x⁶)</strong>, not <strong>11 x³ − 5 x³</strong>. <br>Choice D is incorrect. This is equivalent to <strong>11 x^(6 + 5 x⁶)</strong>, not <strong>11 x³ − 5 x³</strong>.",
    hasFigure: false,
  },
  {
    id: "8452c42b",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>50 x² + 5 x²</strong>?",
    choices: [
      { label: "A", text: "<strong>250 x²</strong>" },
      { label: "B", text: "<strong>10 x²</strong>" },
      { label: "C", text: "<strong>45 x²</strong>" },
      { label: "D", text: "<strong>55 x²</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The given expression shows addition of two like terms. Therefore, the given expression is equivalent to <strong>(50 + 5) x²</strong>, or <strong>55 x²</strong>.<br>Choice A is incorrect. This expression is equivalent to <strong>(50) (5) x²</strong>, not <strong>(50 + 5) x²</strong>.<br>Choice B is incorrect. This expression is equivalent to <strong>((50) / (5)) x²</strong>, not <strong>(50 + 5) x²</strong>.<br>Choice C is incorrect. This expression is equivalent to <strong>(50 − 5) x²</strong>, not <strong>(50 + 5) x²</strong>.",
    hasFigure: false,
  },
  {
    id: "974d33dc",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to the sum of <strong>(r³ + 5 r² + 7, )</strong> and <strong>(r² + 8 r + 12, )</strong> ?",
    choices: [
      {
        label: "A",
        text: "<strong>r to the fifth power + 13 r³ + 19</strong>",
      },
      { label: "B", text: "<strong>2 r³ + 13 r² + 19</strong>" },
      { label: "C", text: "<strong>r³ + 5 r² + 7 r + 12</strong>" },
      { label: "D", text: "<strong>r³ + 6 r² + 8 r + 19</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Grouping like terms, the given expressions can be rewritten as <strong>r³ + (5, r² + r², ) + 8 r + (7 + 12, )</strong>. This can be rewritten as <strong>r³ + 6, r² + 8 r + 19</strong>.Choice A is incorrect and may result from adding the two sets of unlike terms, <strong>r³</strong> and <strong>r²</strong> as well as <strong>5, r²</strong> and <strong>8 r</strong>, and then adding the respective exponents. Choice B is incorrect and may result from adding the unlike terms <strong>r³</strong> and <strong>r²</strong> as if they were <strong>r³</strong> and <strong>r³</strong> and adding the unlike terms <strong>5, r²</strong> and <strong>8 r</strong> as if they were <strong>5, r²</strong> and <strong>8, r²</strong>. Choice C is incorrect and may result from errors when combining like terms.",
    hasFigure: false,
  },
  {
    id: "9ed9f54d",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to <strong>2 · (x² − x, ) + 3 · (x² − x, )</strong> ?",
    choices: [
      { label: "A", text: "<strong>5 x² − 5 x</strong>" },
      { label: "B", text: "<strong>5 x² + 5 x</strong>" },
      { label: "C", text: "5x" },
      { label: "D", text: "5x2" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since <strong>(x² − x, )</strong> is a common term in the original expression, like terms can be added: <strong>2 · (x² − x, ) + 3 · (x² − x, ) = 5 · (x² − x, )</strong>. Distributing the constant term 5 yields <strong>5 x² − 5 x</strong>.Choice B is incorrect and may result from not distributing the negative signs in the expressions within the parentheses. Choice C is incorrect and may result from not distributing the negative signs in the expressions within the parentheses and from incorrectly eliminating the <strong>x²</strong>-term. Choice D is incorrect and may result from incorrectly eliminating the x-term.",
    hasFigure: false,
  },
  {
    id: "beb86a0c",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>9 x² + 5 x</strong>?",
    choices: [
      { label: "A", text: "<strong>x (9 x + 5)</strong>" },
      { label: "B", text: "<strong>5 x (9 x + 1)</strong>" },
      { label: "C", text: "<strong>9 x (x + 5)</strong>" },
      { label: "D", text: "<strong>x² (9 x + 5)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since <strong>x</strong> is a factor of each term in the given expression, the expression is equivalent to <strong>x (9 x) + x (5)</strong>, or <strong>x (9 x + 5)</strong>.<br>Choice B is incorrect. This expression is equivalent to <strong>45 x² + 5 x</strong>, not <strong>9 x² + 5 x</strong>.<br>Choice C is incorrect. This expression is equivalent to <strong>9 x² + 45 x</strong>, not <strong>9 x² + 5 x</strong>.<br>Choice D is incorrect. This expression is equivalent to <strong>9 x³ + 5 x²</strong>, not <strong>9 x² + 5 x</strong>.",
    hasFigure: false,
  },
  {
    id: "d4d513ff",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>12 x + 27</strong>?",
    choices: [
      { label: "A", text: "<strong>12 (9 x + 1)</strong>" },
      { label: "B", text: "<strong>27 (12 x + 1)</strong>" },
      { label: "C", text: "<strong>3 (4 x + 9)</strong>" },
      { label: "D", text: "<strong>3 (9 x + 24)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Each term in the given expression, <strong>12 x + 27</strong>, has a common factor of <strong>3</strong>. Therefore, the expression can be rewritten as <strong>3 (4 x) + 3 (9)</strong>, or <strong>3 (4 x + 9)</strong>. Thus, the expression <strong>3 (4 x + 9)</strong> is equivalent to the given expression.<br>Choice A is incorrect. This expression is equivalent to <strong>108 x + 12</strong>, not <strong>12 x + 27</strong>.<br>Choice B is incorrect. This expression is equivalent to <strong>324 x + 27</strong>, not <strong>12 x + 27</strong>.<br>Choice D is incorrect. This expression is equivalent to <strong>27 x + 72</strong>, not <strong>12 x + 27</strong>.",
    hasFigure: false,
  },
  {
    id: "df0ef054",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the expression above?",
    choices: [
      { label: "A", text: "<strong>x³ + 5 x</strong>" },
      { label: "B", text: "<strong>3 x³ + x</strong>" },
      {
        label: "C",
        text: "<strong>2 x to the sixth power − x to the fourth power − 6 x²</strong>",
      },
      {
        label: "D",
        text: "<strong>3 x to the sixth power − x to the fourth power − 6 x²</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Using the distributive property to multiply the terms in the parentheses yields <strong>(2 x³ · x³, ) + (2 x³ · −2 x, ) + (3 x · x³, ) + (3 x · −2 x, )</strong>, which is equivalent to <strong>2 x to the sixth power − 4 x to the fourth power + 3 x to the fourth power − 6 x²</strong>. Combining like terms results in  <strong>2 x to the sixth power − x to the fourth power − 6 x²</strong>.Choices A and D are incorrect and may result from conceptual errors when multiplying the terms in the given expression. Choice B is incorrect and may result from adding, instead of multiplying, <strong>(2 x³ + 3 x, )</strong> and <strong>(x³ − 2 x, )</strong>.",
    hasFigure: false,
  },
  {
    id: "e312081b",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the given expression?",
    choices: [
      { label: "A", text: "<strong>3 x − 2</strong>" },
      { label: "B", text: "<strong>3 x + 2</strong>" },
      { label: "C", text: "<strong>3 x − 8</strong>" },
      { label: "D", text: "<strong>3 x + 8</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Using the associative and commutative properties of addition, the given expression <strong>(x + 5, ) + (2 x − 3, )</strong> can be rewritten as <strong>(x + 2 x, ) + (5 − 3, )</strong>. Adding these like terms results in <strong>3 x + 2</strong>.Choice A is incorrect and may result from adding <strong>(x − 5, ) + (2 x + 3, )</strong>. Choice C is incorrect and may result from adding <strong>(x − 5, ) + (2 x − 3, )</strong>. Choice D is incorrect and may result from adding <strong>(x + 5, ) + (2 x + 3, )</strong>.",
    hasFigure: false,
  },
  {
    id: "e597050f",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>9 x + 6 x + 2 y + 3 y</strong>?",
    choices: [
      { label: "A", text: "<strong>3 x + 5 y</strong>" },
      { label: "B", text: "<strong>6 x + 8 y</strong>" },
      { label: "C", text: "<strong>12 x + 8 y</strong>" },
      { label: "D", text: "<strong>15 x + 5 y</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Combining like terms in the given expression yields <strong>(9 x + 6 x) + (2 y + 3 y)</strong>, or <strong>15 x + 5 y</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "f5c3e3b8",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(m⁴ q⁴ z^(−1)) (m q⁵ z³)</strong>, where <strong>m</strong>, <strong>q</strong>, and <strong>z</strong> are positive?",
    choices: [
      { label: "A", text: "<strong>m⁴ q²⁰ z^(−3)</strong>" },
      { label: "B", text: "<strong>m⁵ q⁹ z²</strong>" },
      { label: "C", text: "<strong>m⁶ q⁸ z^(−1)</strong>" },
      { label: "D", text: "<strong>m²⁰ q¹² z^(−2)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Applying the commutative property of multiplication, the expression <strong>(m⁴ q⁴ z^(−1)) (m q⁵ z³)</strong> can be rewritten as <strong>(m⁴ m) (q⁴ q⁵) (z^(−1) z³)</strong>. For positive values of <strong>x</strong>, <strong>(x^(a)) (x^(b)) = x^(a + b)</strong>. Therefore, the expression <strong>(m⁴ m) (q⁴ q⁵) (z^(−1) z³)</strong> can be rewritten as <strong>(m^(4 + 1)) (q^(4 + 5)) (z^(−1 + 3))</strong>, or <strong>m⁵ q⁹ z²</strong>.<br>Choice A is incorrect and may result from multiplying, not adding, the exponents.<br>Choice C is incorrect and may result from conceptual or calculation errors. <br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "fb96a5b3",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to <strong>2 · (a, b − 3, ) + 2</strong> ?",
    choices: [
      { label: "A", text: "<strong>2 a, b − 1</strong>" },
      { label: "B", text: "<strong>2 a, b − 4</strong>" },
      { label: "C", text: "<strong>2 a, b − 5</strong>" },
      { label: "D", text: "<strong>2 a, b − 8</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Applying the distributive property to the given expression yields <strong>2 · a, b + 2 · −3 + 2</strong>, or <strong>2 a, b − 6 + 2</strong>. Adding the like terms <strong>−6</strong> and 2 results in the expression <strong>2 a, b − 4</strong>.Choice A is incorrect and may result from multiplying <strong>a, b</strong> by 2 without multiplying <strong>−3</strong> by 2 when applying the distributive property. Choices C and D are incorrect and may result from computational or conceptual errors.",
    hasFigure: false,
  },
  {
    id: "fd4b2aa0",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>12 x³ − 5 x³</strong>?",
    choices: [
      { label: "A", text: "<strong>7 x⁶</strong>" },
      { label: "B", text: "<strong>17 x³</strong>" },
      { label: "C", text: "<strong>7 x³</strong>" },
      { label: "D", text: "<strong>17 x⁶</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The given expression shows subtraction of two like terms. The two terms can be subtracted as follows: <strong>12 x³ − 5 x³ = (12 − 5) x³</strong>, or <strong>7 x³</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the result of adding, not subtracting, the two like terms.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "fd65f47f",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(2 x² + x − 9) + (x² + 6 x + 1)</strong>?",
    choices: [
      { label: "A", text: "<strong>2 x² + 7 x + 10</strong>" },
      { label: "B", text: "<strong>2 x² + 6 x − 8</strong>" },
      { label: "C", text: "<strong>3 x² + 7 x − 10</strong>" },
      { label: "D", text: "<strong>3 x² + 7 x − 8</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The given expression is equivalent to <strong>(2 x² + x + (−9)) + (x² + 6 x + 1)</strong>, which can be rewritten as <strong>(2 x² + x²) + (x + 6 x) + (−9 + 1)</strong>. Adding like terms in this expression yields <strong>3 x² + 7 x + (−8)</strong>, or <strong>3 x² + 7 x − 8</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];

export const EQUIV_EXPR_MEDIUM: PracticeQuestion[] = [
  {
    id: "0b3d25c5",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to <strong>the fourth root of, x² + 8 x + 16, end root</strong>, where <strong>x > 0</strong>?",
    choices: [
      { label: "A", text: "<strong>(x + 4, ), to the fourth power</strong>" },
      { label: "B", text: "<strong>(x + 4, ), ²</strong>" },
      { label: "C", text: "<strong>(x + 4, )</strong>" },
      {
        label: "D",
        text: "<strong>(x + 4, ), raised to the one half power</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The given expression can also be written as <strong>(x² + 8, x + 16, ), raised to the one fourth power</strong>. The trinomial <strong>x² + 8, x + 16</strong> can be rewritten in factored form as <strong>(x + 4, ), ²</strong>. Thus, the entire expression can be rewritten as <strong>((x + 4, ), ², ), raised to the one fourth power</strong>. Simplifying the exponents yields <strong>(x + 4, ), raised to the one half power</strong>.Choices A, B, and C are incorrect and may result from errors made when simplifying the exponents in the expression <strong>((x + 4, ), ², ), raised to the one fourth power</strong>.",
    hasFigure: false,
  },
  {
    id: "16de54c7",
    type: "spr",
    questionHtml:
      "If the given expression is rewritten in the form <strong>(2 x − 3, ) · (x + k, )</strong>, where k is a constant, what is the value of k ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 4. It’s given that <strong>2 x² + 5 x − 12</strong> can be rewritten as <strong>(2 x − 3, ) · (x + k, )</strong>; it follows that <strong>(2 x − 3, ) · (x + k, ) = 2 x² + 5 x − 12</strong>. Expanding the left-hand side of this equation yields <strong>2 x² + 2 k x − 3 x − 3 k = 2 x² + 5 x − 12</strong>. Subtracting <strong>2 x²</strong> from both sides of this equation yields <strong>2 k x − 3 x − 3 k = 5 x − 12</strong>. Using properties of equality, <strong>2 k x − 3 x = 5 x</strong> and <strong>−3 k = −12</strong>. Either equation can be solved for k. Dividing both sides of <strong>−3 k = −12</strong> by <strong>−3</strong> yields <strong>k = 4</strong>. The equation <strong>2 k x − 3 x = 5 x</strong> can be rewritten as <strong>x · (2 k − 3, ) = 5 x</strong>. It follows that <strong>2 k − 3 = 5</strong>. Solving this equation for k also yields <strong>k = 4</strong>. Therefore, the value of k is 4.",
    hasFigure: false,
  },
  {
    id: "1dd13816",
    type: "spr",
    questionHtml:
      "<strong>(5 x³ − 3) − (−4 x³ + 8)</strong><br>The given expression is equivalent to <strong>b x³ − 11</strong>, where <strong>b</strong> is a constant. What is the value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "9",
    explanation:
      "The correct answer is <strong>9</strong>. The given expression can be rewritten as <strong>(5 x³ − 3) + (−1) (−4 x³ + 8)</strong>. By applying the distributive property, this expression can be rewritten as <strong>5 x³ − 3 + 4 x³ + (−8)</strong>, which is equivalent to <strong>(5 x³ + 4 x³) + (−3 + (−8))</strong>. Adding like terms in this expression yields <strong>9 x³ − 11</strong>. Since it's given that <strong>(5 x³ − 3) − (−4 x³ + 8)</strong> is equivalent to <strong>b x³ − 11</strong>, it follows that <strong>9 x³ − 11</strong> is equivalent to <strong>b x³ − 11</strong>. Therefore, the coefficients of <strong>x³</strong> in these two expressions must be equivalent, and the value of <strong>b</strong> must be <strong>9</strong>.",
    hasFigure: false,
  },
  {
    id: "26eb61c1",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>6 x⁸ y² + 12 x² y²</strong>?",
    choices: [
      { label: "A", text: "<strong>6 x² y² (2 x⁶)</strong>" },
      { label: "B", text: "<strong>6 x² y² (x⁴)</strong>" },
      { label: "C", text: "<strong>6 x² y² (x⁶ + 2)</strong>" },
      { label: "D", text: "<strong>6 x² y² (x⁴ + 2)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since each term of the given expression has a common factor of <strong>6 x² y²</strong>, it may be rewritten as <strong>6 x² y² (x⁶) + 6 x² y² (2)</strong>, or <strong>6 x² y² (x⁶ + 2)</strong>.<br>Choice A is incorrect. This expression is equivalent to <strong>12 x⁸ y²</strong>, not <strong>6 x⁸ y² + 12 x² y²</strong>.<br>Choice B is incorrect. This expression is equivalent to <strong>6 x⁶ y²</strong>, not <strong>6 x⁸ y² + 12 x² y²</strong>.<br>Choice D is incorrect. This expression is equivalent to <strong>6 x⁶ y² + 12 x² y²</strong>, not <strong>6 x⁸ y² + 12 x² y²</strong>.",
    hasFigure: false,
  },
  {
    id: "3e9cc0c2",
    type: "mcq",
    questionHtml: "Which of the following is equivalent to [] ?",
    choices: [
      { label: "A", text: "<strong>1 − p to the power 8</strong>" },
      { label: "B", text: "<strong>1 − p to the power 7</strong>" },
      { label: "C", text: "<strong>1 − p to the power 6</strong>" },
      { label: "D", text: "<strong>1 − p to the power 5</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Multiplying (1 – p) by each term of the polynomial within the second pair of parentheses gives (1 – p)1 = 1 – p; (1 – p)p = p – p2; (1 – p)p2 = p2 – p3; (1 – p)p3 = p3 – p4; (1 – p)p4 = p4 – p5; (1 – p)p5 = p5 – p6; and (1 – p)p6 = p6 – p7. Adding these seven expressions together and combining like terms gives 1 + (p – p) + (p2 – p2) + (p3 – p3) + (p4 – p4) + (p5 – p5) + (p6 – p6) – p7, which can be simplified to 1 – p7.<br><br> Choices A, C, and D are incorrect and may result from incorrectly identifying the highest power of p in the expressions or incorrectly combining like terms.",
    hasFigure: false,
  },
  {
    id: "42c71eb5",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the expression above?",
    choices: [
      { label: "A", text: "<strong>4 x² + 21 x + 33</strong>" },
      { label: "B", text: "<strong>4 x² + 21 x + 29</strong>" },
      { label: "C", text: "<strong>4 x² + x + 29</strong>" },
      { label: "D", text: "<strong>4 x² + x + 33</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The given expression can be rewritten as <strong>(2 x + 5, ), ² + −1 · (x − 2, ) + 2 · (x + 3, )</strong>. Applying the distributive property, the expression  <strong>−1 · (x − 2, ) + 2 · (x + 3, )</strong>can be rewritten as <strong>−one · x + −1 · −2 + 2 · x + 2 · 3</strong>, or <strong>−x + 2 + 2 x + 6</strong>. Adding like terms yields <strong>x + 8</strong>. Substituting <strong>x + 8</strong> for  <strong>−1 · (x − 2, ) + 2 · (x + 3, )</strong> in the given expression yields <strong>(2 x + 5, ), ² + x + 8</strong>. By the rules of exponents, the expression <strong>(2 x + 5, ), ²</strong> is equivalent to <strong>(2 x + 5, ) · (2 x + 5, )</strong>. Applying the distributive property, this expression can be rewritten as <strong>2 x · 2x + 2 x · 5 + 5 · 2 x + 5 · 5</strong>, or <strong>4 x² + 10 x + 10 x + 25</strong>. Adding like terms gives <strong>4 x² + 20 x + 25</strong>. Substituting <strong>4 x² + 20 x + 25</strong> for <strong>(2 x + 5, ), ²</strong> in the rewritten expression yields <strong>4 x² + 20 x + 25 + x + 8</strong>, and adding like terms yields <strong>4 x² + 21 x + 33</strong>.Choices B, C, and D are incorrect. Choices C and D may result from rewriting the expression <strong>(2 x + 5, ), ²</strong> as <strong>4 x² + 25</strong>, instead of as <strong>4 x² + 20 x + 25</strong>. Choices B and C may result from rewriting the expression <strong>negative, (x − 2, )</strong> as <strong>−x − 2</strong>, instead of <strong>−x + 2</strong>.",
    hasFigure: false,
  },
  {
    id: "42f19012",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>a^((11) / (12))</strong>, where <strong>a > 0</strong>?",
    choices: [
      { label: "A", text: "<strong>RootIndex 12 √(a¹³²)</strong>" },
      { label: "B", text: "<strong>RootIndex 144 √(a¹³²)</strong>" },
      { label: "C", text: "<strong>RootIndex 121 √(a¹³²)</strong>" },
      { label: "D", text: "<strong>RootIndex 11 √(a¹³²)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since <strong>(12) / (12) = 1</strong>, multiplying the exponent of the given expression by <strong>(12) / (12)</strong> yields an equivalent expression: <strong>a^(((11) / (12)) ((12) / (12))) = a^(((132) / (144)))</strong>. Since <strong>(132) / (144) = 132 ((1) / (144))</strong>, the expression <strong>a^((132) / (144))</strong> can be rewritten as <strong>a^((132) ((1) / (144)))</strong>. Applying properties of exponents, this expression can be rewritten as <strong>(a¹³²)^((1) / (144))</strong>. An expression of the form <strong>(m)^((1) / (k))</strong>, where <strong>m > 0</strong> and <strong>k > 0</strong>, is equivalent to <strong>RootIndex k √(m)</strong>. Therefore, <strong>(a¹³²)^((1) / (144))</strong> is equivalent to <strong>RootIndex 144 √(a¹³²)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors. <br>Choice C is incorrect and may result from conceptual or calculation errors. <br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "463eec13",
    type: "mcq",
    questionHtml:
      "If <strong>x ≠ 0</strong>, which of the following expressions is equivalent to <strong>the fraction with numerator the √ 16 x to the fourth power, y to the eighth power, end root, and denominator x³, end fraction</strong> ?",
    choices: [
      { label: "A", text: "<strong>8 x², y to the fourth power</strong>" },
      { label: "B", text: "<strong>4 x, y to the fourth power</strong>" },
      { label: "C", text: "<strong>4 x raised to the − 2 power, y²</strong>" },
      {
        label: "D",
        text: "<strong>4 x raised to the − 1 power, y to the fourth power</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Taking the square root of an exponential expression halves the exponent, so <strong>the fraction with numerator the √, 16 x to the fourth power · y to the eighth power, end root, and denominator x³, end fraction = the fraction with numerator 4 x² · y to the fourth power, and denominator x³</strong>, which further reduces to <strong>the fraction with numerator 4, y to the fourth power, and denominator x</strong>. This can be rewritten as <strong>4, x to the − 1 power · y to the fourth power</strong>.Choice A is incorrect and may result from neglecting the denominator of the given expression and from incorrectly calculating the square root of 16. Choice B is incorrect and may result from rewriting <strong>1 over x</strong> as <strong>x to the first power</strong> rather than <strong>x to the − 1 power</strong>. Choice C is incorrect and may result from taking the square root of the variables in the numerator twice instead of once.",
    hasFigure: false,
  },
  {
    id: "4eaf0a3a",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>RootIndex 7 √(x⁹ y⁹)</strong>, where <strong>x</strong> and <strong>y</strong> are positive?",
    choices: [
      { label: "A", text: "<strong>(xy)^(seven ninths)</strong>" },
      { label: "B", text: "<strong>(xy)^(nine sevenths)</strong>" },
      { label: "C", text: "<strong>(xy)¹⁶</strong>" },
      { label: "D", text: "<strong>(xy)⁶³</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. For positive values of <strong>a</strong> and <strong>b</strong>, <strong>a^(m) b^(m) = (a b)^(m)</strong>, <strong>RootIndex n √(a) = (a)^((1) / (n))</strong>, and <strong>(a^(j))^(k) = a^(j k)</strong>. Therefore, the given expression, <strong>RootIndex 7 √(x⁹ y⁹)</strong>, can be rewritten as <strong>RootIndex 7 √((xy)⁹)</strong>. This expression is equivalent to <strong>((xy)⁹)^(one seventh)</strong>, which can be rewritten as <strong>(xy)^(9 dot one seventh)</strong>, or <strong>(xy)^(nine sevenths)</strong>. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "52931bfa",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(8 x (x − 7) − 3 (x − 7)) / (2 x − 14)</strong>, where <strong>x > 7</strong>?",
    choices: [
      { label: "A", text: "<strong>(x − 7) / (5)</strong>" },
      { label: "B", text: "<strong>(8 x − 3) / (2)</strong>" },
      { label: "C", text: "<strong>(8 x² − 3 x − 14) / (2 x − 14)</strong>" },
      { label: "D", text: "<strong>(8 x² − 3 x − 77) / (2 x − 14)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The given expression has a common factor of <strong>2</strong> in the denominator, so the expression can be rewritten as <strong>(8 x (x − 7) − 3 (x − 7)) / (2 (x − 7))</strong>. The three terms in this expression have a common factor of <strong>(x − 7)</strong>. Since it's given that <strong>x > 7</strong>, <strong>x</strong> can't be equal to <strong>7</strong>, which means <strong>(x − 7)</strong> can't be equal to <strong>0</strong>. Therefore, each term in the expression, <strong>(8 x (x − 7) − 3 (x − 7)) / (2 (x − 7))</strong>, can be divided by <strong>(x − 7)</strong>, which gives <strong>(8 x − 3) / (2)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "5805e747",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(7 x³ + 7 x) − (6 x³ − 3 x)</strong>?",
    choices: [
      { label: "A", text: "<strong>x³ + 10 x</strong>" },
      { label: "B", text: "<strong>− 13 x³ + 10 x</strong>" },
      { label: "C", text: "<strong>− 13 x³ + 4 x</strong>" },
      { label: "D", text: "<strong>x³ + 4 x</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Applying the distributive property, the given expression can be written as <strong>7 x³ + 7 x − 6 x³ + 3 x</strong>. Grouping like terms in this expression yields <strong>(7 x³ − 6 x³) + (7 x + 3 x)</strong>. Combining like terms in this expression yields <strong>x³ + 10 x</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "5b6af6b1",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(d − 6) (8 d² − 3)</strong>?",
    choices: [
      { label: "A", text: "<strong>8 d³ − 14 d² − 3 d + 18</strong>" },
      { label: "B", text: "<strong>8 d³ − 17 d² + 48</strong>" },
      { label: "C", text: "<strong>8 d³ − 48 d² − 3 d + 18</strong>" },
      { label: "D", text: "<strong>8 d³ − 51 d² + 48</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Applying the distributive property to the given expression yields <strong>d (8 d² − 3) − 6 (8 d² − 3)</strong>. Applying the distributive property once again to this expression yields <strong>(d) (8 d²) + (d) (−3) + (−6) (8 d²) + (−6) (−3)</strong>, or <strong>8 d³ + (−3 d) + (−48 d²) + 18</strong>. This expression can be rewritten as <strong>8 d³ − 48 d² − 3 d + 18</strong>. Thus, <strong>(d − 6) (8 d² − 3)</strong> is equivalent to <strong>8 d³ − 48 d² − 3 d + 18</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "6d04c89d",
    type: "mcq",
    questionHtml:
      "The expression <strong>(24) / (6 x + 42)</strong> is equivalent to <strong>(4) / (x + b)</strong>, where b is a constant and <strong>x > 0</strong>. What is the value of b?",
    choices: [
      { label: "A", text: "<strong>7</strong>" },
      { label: "B", text: "<strong>10</strong>" },
      { label: "C", text: "<strong>24</strong>" },
      { label: "D", text: "<strong>252</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since the given expressions are equivalent and the numerator of the second expression is <strong>one sixth</strong> of the numerator of the first expression, the denominator of the second expression must also be <strong>one sixth</strong> of the denominator of the first expression. By the distributive property,  <strong>one sixth (6 x + 42)</strong> is equivalent to <strong>one sixth (6 x) + one sixth (42)</strong>, or <strong>x + 7</strong>. Therefore, the value of <strong>b</strong> is <strong>7</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "7348f046",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the given expression?",
    choices: [
      { label: "A", text: "<strong>x − 4</strong>" },
      { label: "B", text: "<strong>3 x − 4</strong>" },
      { label: "C", text: "<strong>x + 10</strong>" },
      { label: "D", text: "<strong>2 x² + 21</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Distributing the negative sign to the terms in the second parentheses yields <strong>(2 x + 3, ) − x + 7</strong>. This expression can be rewritten as <strong>2 x − x + 3 + 7</strong>. Combining like terms results in <strong>x + 10</strong>.Choice A is incorrect and may result from not distributing the negative sign to the 7. Choice B is incorrect and may result from adding <strong>x − 7</strong> to <strong>2 x + 3</strong> instead of subtracting <strong>x − 7</strong>. Choice D is incorrect and may result from adding the product of <strong>2 x</strong> and x to the product of 3 and 7.",
    hasFigure: false,
  },
  {
    id: "8838a672",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to the expression above?",
    choices: [
      { label: "A", text: "<strong>−10, x³ − 3, x² + x + 3</strong>" },
      { label: "B", text: "<strong>−2, x³ − 7, x² + x + 3</strong>" },
      { label: "C", text: "<strong>−2, x³ − 3, x² + x + 3</strong>" },
      { label: "D", text: "<strong>10, x³ − 7, x² − x + 3</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Using the distributive property, the given expression can be rewritten as <strong>4 x³ − 5 x² + 3 − 6 x³ − 2 x² + x</strong>. Combining like terms, this expression can be rewritten as <strong>(4 − 6, ) · x³ + (−5 − 2, ) · x² + x + 3</strong>, which is equivalent to <strong>−2 x³ − 7 x² + x + 3</strong>.Choices A, C, and D are incorrect and may result from an error when applying the distributive property or an error when combining like terms.",
    hasFigure: false,
  },
  {
    id: "a05bd3a4",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to <strong>x² − 5</strong> ?",
    choices: [
      { label: "A", text: "<strong>(x + the √ 5, ), ²</strong>" },
      { label: "B", text: "<strong>(x − the √ 5, ), ²</strong>" },
      {
        label: "C",
        text: "<strong>(x + the √ 5, ) · (x − the √ 5, )</strong>",
      },
      { label: "D", text: "<strong>(x + 5, ) · (x − 1, )</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The expression can be written as a difference of squares x2 – y2, which can be factored as (x + y)(x – y). Here, y2 = 5, so <strong>y = the √ 5</strong>, and the expression therefore factors as <strong>(x + the √ 5, ) · (x − the √ 5, )</strong>.Choices A and B are incorrect and may result from misunderstanding how to factor a difference of squares. Choice D is incorrect; (x + 5)(x – 1) can be rewritten as x2 + 4x – 5, which is not equivalent to the original expression.",
    hasFigure: false,
  },
  {
    id: "a1bf1c4e",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the expression above?",
    choices: [
      { label: "A", text: "(x + 3)2 + 5" },
      { label: "B", text: "(x + 3)2 – 5" },
      { label: "C", text: "(x – 3)2 + 5" },
      { label: "D", text: "(x – 3)2 – 5" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The given quadratic expression is in standard form, and each answer choice is in vertex form. Completing the square converts the expression from standard form to vertex form. The first step is to rewrite the expression as follows: <strong>x² + 6 x + 4 = x² + 6 x + 9 + 4 − 9</strong>. The first three terms of the revised expression can be rewritten as a perfect square as follows: <strong>x² + 6 x + 9 + 4 − 9 = (x + 3, ), ² + 4 − 9</strong>. Combining the constant terms gives <strong>(x + 3, ), ² − 5</strong>.Choice A is incorrect. Squaring the binomial and simplifying the expression in choice A gives <strong>x² + 6 x + 9 + 5</strong>. Combining like terms gives <strong>x² − 6 x + 14</strong>, not <strong>x² + 6 x + 4</strong>. Choice C is incorrect. Squaring the binomial and simplifying the expression in choice C gives <strong>x² − 6 x + 9 + 5</strong>. Combining like terms gives <strong>x² − 6 x + 14</strong>, not <strong>x² + 6 x + 4</strong>. Choice D is incorrect. Squaring the binomial and simplifying the expression in choice D gives <strong>x² − 6 x + 9 − 5</strong>. Combining like terms gives <strong>x² − 6 x + 4</strong>, not <strong>x² + 6 x + 4</strong>.",
    hasFigure: false,
  },
  {
    id: "a255ae72",
    type: "mcq",
    questionHtml:
      "If <strong>x² = a + b</strong> and <strong>y² = a + c</strong>, which of the following is equal to <strong>(x² − y², ), ²</strong> ?",
    choices: [
      { label: "A", text: "<strong>a, ² − 2 a, c + c²</strong>" },
      { label: "B", text: "<strong>b² − 2 b c + c²</strong>" },
      { label: "C", text: "<strong>4, a, ² − 4 a, b c + c²</strong>" },
      { label: "D", text: "<strong>4 a, ² − 2 a, b c + b² c²</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that <strong>x² = a + b</strong> and <strong>y² = a + c</strong>. Using the distributive property, the expression <strong>(x² − y², ), ²</strong> can be rewritten as <strong>(x², ), ² − 2 x² · y² + (y², ), ²</strong>. Substituting <strong>a + b</strong> and <strong>a + c</strong> for <strong>x²</strong> and <strong>y²</strong>, respectively, in this expression yields <strong>(a + b, ), ² − 2 · ((a + b, ) · (a + c, ), ) + (a + c, ), ²</strong>. Expanding this expression yields <strong>(a, ² + 2 a, b + b², ) − (2 a, ² + 2 b c + 2 a, c + 2 a, b, ) + (a, ² + 2 a, c + c², )</strong>. Combining like terms, this expression can be rewritten as <strong>b² − 2 b c + c²</strong>.Choices A, C, and D are incorrect and may result from an error in using the distributive property, substituting, or combining like terms.",
    hasFigure: false,
  },
  {
    id: "a391ed22",
    type: "spr",
    questionHtml:
      "The expression above is equivalent to <br> <strong>a, x² + b x + c</strong>, where a, b, and c are constants. What is the value of b?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is <strong>five halves</strong>. The expression <strong>(one half x + three halves, ) · (three halves x + one half, )</strong> can be written in the form <strong>a, x² + b x + c</strong>, where a, b, and c are constants, by multiplying out the expression using the distributive property of multiplication over addition. The result is <strong>one half x · three halves x + one half x · one half + three halves · three halves x + three halves · one half</strong>. This expression can be rewritten by multiplying as indicated to give <strong>three fourths x² + one fourth x + nine fourths x + three fourths</strong>, which can be simplified to <strong>three fourths x² + ten fourths x + three fourths</strong>, or <strong>three fourths x² + five halves x + three fourths</strong>. This is in the form <strong>a, x² + b x + c</strong>, where the value of b is <strong>five halves</strong>. Note that 5/2 and 2.5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "a520ba07",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to the expression above?",
    choices: [
      { label: "A", text: "<strong>y²</strong>" },
      { label: "B", text: "<strong>x · y²</strong>" },
      { label: "C", text: "<strong>y³</strong>" },
      { label: "D", text: "<strong>x · y³</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. One of the properties of radicals is <strong>the nth root of a · b, end root = the nth root of a, end root · the nth root of b, end root</strong>. Thus, the given expression can be rewritten as <strong>the cube root of x³, end root · the cube root of y to the power 6, end root</strong>. Simplifying by taking the cube root of each part gives x1 ⋅ y2, or xy2.Choices A, C, and D are incorrect and may be the result of incorrect application of the properties of exponents and radicals.",
    hasFigure: false,
  },
  {
    id: "ad2ec615",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the expression <strong>x to the fourth power − x² − 6</strong> ?",
    choices: [
      { label: "A", text: "<strong>(x² + 1, ) · (x² − 6, )</strong>" },
      { label: "B", text: "<strong>(x² + 2, ) · (x² − 3, )</strong>" },
      { label: "C", text: "<strong>(x² + 3, ) · (x² − 2, )</strong>" },
      { label: "D", text: "<strong>(x² + 6, ) · (x² − 1, )</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The term x4 can be factored as <strong>x² · x²</strong>. Factoring –6 as <strong>2 · −3</strong> yields values that add to –1, the coefficient of x2 in the expression.Choices A, C, and D are incorrect and may result from finding factors of –6 that don’t add to the coefficient of x2 in the original expression.",
    hasFigure: false,
  },
  {
    id: "b47419f4",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the expression above?",
    choices: [
      { label: "A", text: "<strong>−one sixth x + 8</strong>" },
      { label: "B", text: "<strong>−one sixth x − 2</strong>" },
      { label: "C", text: "<strong>−one-third x² + one-half x + 15</strong>" },
      {
        label: "D",
        text: "<strong>−one-third x² − nine halves x − 15</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. By distributing the minus sign through the expression <strong>two thirds x − 5</strong>, the given expression can be rewritten as <strong>(one half x + 3, ) − two thirds x + 5</strong>, which is equivalent to <strong>one half x − two thirds x + 3 + 5</strong>. Combining like terms gives <strong>(one half − two thirds, ) · x + (3 + 5, )</strong>, or <strong>−one sixth x + 8</strong>.Choice B is incorrect and may be the result of failing to distribute the minus sign appropriately through the second term and simplifying the expression <strong>one half x + 3 − two thirds x − 5</strong>. Choice C is incorrect and may be the result of multiplying the expressions <strong>one half x + 3</strong> and <strong>−two thirds x + 5</strong>. Choice D is incorrect and may be the result of multiplying the expressions <strong>one half x + 3</strong> and <strong>−two thirds x − 5</strong>.",
    hasFigure: false,
  },
  {
    id: "b4a6ed81",
    type: "spr",
    questionHtml:
      "The expression <strong>90 y^(5 − 54 y⁴)</strong> is equivalent to <strong>r y⁴ (15 y − 9)</strong>, where <strong>r</strong> is a constant. What is the value of <strong>r</strong>?",
    choices: [],
    correctAnswer: "6",
    explanation:
      "The correct answer is <strong>6</strong>. Applying the distributive property to the expression <strong>r y⁴ (15 y − 9)</strong> yields <strong>15 r y^(5 − 9 r y⁴)</strong>. Since <strong>90 y^(5 − 54 y⁴)</strong> is equivalent to <strong>r y⁴ (15 y − 9)</strong>, it follows that <strong>90 y^(5 − 54 y⁴)</strong> is also equivalent to <strong>15 r y^(5 − 9 r y⁴)</strong>. Since these expressions are equivalent, it follows that corresponding coefficients are equivalent. Therefore, <strong>90 = 15 r</strong> and <strong>−54 = − 9 r</strong>. Solving either of these equations for <strong>r</strong> will yield the value of <strong>r</strong>. Dividing both sides of <strong>90 = 15 r</strong> by <strong>15</strong> yields <strong>6 = r</strong>. Therefore, the value of <strong>r</strong> is <strong>6</strong>.",
    hasFigure: false,
  },
  {
    id: "b8caaf84",
    type: "mcq",
    questionHtml:
      "If <strong>p = 3 x + 4</strong> and <strong>v = x + 5</strong>, which of the following is equivalent to <strong>p v − 2 p + v</strong> ?",
    choices: [
      { label: "A", text: "<strong>3 x² + 12 x + 7</strong>" },
      { label: "B", text: "<strong>3 x² + 14 x + 17</strong>" },
      { label: "C", text: "<strong>3 x² + 19 x + 20</strong>" },
      { label: "D", text: "<strong>3 x² + 26 x + 33</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that <strong>p = 3 x + 4</strong> and <strong>v = x + 5</strong>. Substituting the values for p and v into the expression <strong>p v − 2 p + v</strong> yields <strong>(3 x + 4, ) · (x + 5, ) − 2 · (3 x + 4, ) + x + 5</strong>. Multiplying the terms <strong>(3 x + 4, ) · (x + 5, )</strong> yields <strong>3 x² + 4 x + 15 x + 20</strong>. Using the distributive property to rewrite <strong>−2 · (3 x + 4, )</strong> yields <strong>−6 x − 8</strong>. Therefore, the entire expression can be represented as <strong>3 x² + 4 x + 15 x + 20 − 6 x − 8 + x + 5</strong>. Combining like terms yields <strong>3 x² + 14 x + 17</strong>.Choice A is incorrect and may result from subtracting, instead of adding, the term <strong>x + 5</strong>. Choice C is incorrect. This is the result of multiplying the terms <strong>(3 x + 4, ) · (x + 5, )</strong>. Choice D is incorrect and may result from distributing 2, instead of <strong>−2</strong>, to the term <strong>3 x + 4</strong>.",
    hasFigure: false,
  },
  {
    id: "c3a72da5",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the sum of <strong>3 x to the fourth power + 2 x³</strong> and <strong>4 x to the fourth power + 7 x³</strong>?",
    choices: [
      { label: "A", text: "<strong>16 x to the fourteenth power</strong>" },
      {
        label: "B",
        text: "<strong>7 x to the eighth power + 9 x to the sixth power</strong>",
      },
      { label: "C", text: "<strong>12 x to the fourth power + 14 x³</strong>" },
      { label: "D", text: "<strong>7 x to the fourth power + 9 x³</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Adding the two expressions yields <strong>3 x to the fourth power + 2 x to the third power + 4 x to the fourth power + 7 x to the third power</strong>. Because the pair of terms <strong>3 x to the fourth power</strong> and <strong>4 x to the fourth power</strong> and the pair of terms <strong>2 x to the third power</strong> and <strong>7 x to the third power</strong> each contain the same variable raised to the same power, they are like terms and can be combined as <strong>7 x to the fourth power</strong> and <strong>9 x to the third power</strong>, respectively. The sum of the given expressions therefore simplifies to <strong>7 x to the fourth power + 9 x to the third power</strong>.Choice A is incorrect and may result from adding the coefficients and the exponents in the given expressions. Choice B is incorrect and may result from adding the exponents as well as the coefficients of the like terms in the given expressions. Choice C is incorrect and may result from multiplying, rather than adding, the coefficients of the like terms in the given expressions.",
    hasFigure: false,
  },
  {
    id: "c602140f",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to the expression above?",
    choices: [
      { label: "A", text: "<strong>x − 23 y</strong>" },
      { label: "B", text: "<strong>2 x² − xy − 3 y²</strong>" },
      { label: "C", text: "<strong>2 x² + 24 xy + 36 y²</strong>" },
      { label: "D", text: "<strong>2 x² − 49 xy + 69 y²</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Expanding all terms yields (x – 11y)(2x – 3y) – 12y(–2x + 3y), which is equivalent to 2x2 – 22xy – 3xy + 33y2 + 24xy – 36y2. Combining like terms gives 2x2 – xy – 3y2.Choice A is incorrect and may be the result of using the sums of the coefficients of the existing x and y terms as the coefficients of the x and y terms in the new expressions. Choice C is incorrect and may be the result of incorrectly combining like terms. Choice D is incorrect and may be the result of using the incorrect sign in front of the 12y term.",
    hasFigure: false,
  },
  {
    id: "cc776a04",
    type: "mcq",
    questionHtml:
      "Which of the following is an equivalent form of <strong>(1 . 5 x − 2 . 4, ), ² − (5 . 2, x² − 6 . 4, )</strong> ?",
    choices: [
      { label: "A", text: "<strong>−2 . 2, x² + 1 . 6</strong>" },
      { label: "B", text: "<strong>−2 . 2, x² + 11 . 2</strong>" },
      { label: "C", text: "<strong>−2 . 95, x² − 7 . 2 x + 12 . 16</strong>" },
      { label: "D", text: "<strong>−2 . 95, x² − 7 . 2 x + 0 . 64</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The first expression <strong>(1 . 5 x − 2 . 4, ), ²</strong> can be rewritten as <strong>(1 . 5 x − 2 . 4, ) · (1 . 5 x − 2 . 4, )</strong> . Applying the distributive property to this product yields <strong>(2 . 2 5, x² − 3 . 6 x − 3 . 6 x + 5 . 7 6, ) − (5 . 2, x² − 6 . 4, )</strong> . This difference can be rewritten as <strong>(2 . 2 5, x² − 3 . 6 x − 3 . 6 x + 5 . 7 6, ) + −1 · (5 . 2 x² − 6 . 4, )</strong> . Distributing the factor of <strong>−1</strong> through the second expression yields <strong>2 . 2 5 x, ² − 3 . 6 x − 3 . 6 x + 5 . 7 6 − 5 . 2, x² + 6 . 4</strong> . Regrouping like terms, the expression becomes <strong>(2 . 2 5, x² − 5 . 2, x², ) + (−3 . 6 x − 3 . 6 x, ) + (5 . 7 6 + 6 . 4, )</strong> . Combining like terms yields <strong>−2 . 9 5, x² − 7 . 2 x + 12 . 1 6</strong> .Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.",
    hasFigure: false,
  },
  {
    id: "d9137a84",
    type: "mcq",
    questionHtml:
      "Which expression represents the product of <strong>(x^(−6) y³ z⁵)</strong> and <strong>(x⁴ z⁵ + y⁸ z^(−7))</strong>?",
    choices: [
      { label: "A", text: "<strong>x^(−2) z^(10 + y¹¹) z^(−2)</strong>" },
      { label: "B", text: "<strong>x^(−2) z^(10 + x^(−6)) z^(−2)</strong>" },
      { label: "C", text: "<strong>x^(−2) y³ z^(10 + y⁸) z^(−7)</strong>" },
      {
        label: "D",
        text: "<strong>x^(−2) y³ z^(10 + x^(−6)) y¹¹ z^(−2)</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The product of <strong>(x^(−6) y³ z⁵)</strong> and <strong>(x⁴ z⁵ + y⁸ z^(−7))</strong> can be represented by the expression <strong>(x^(−6) y³ z⁵) (x⁴ z⁵ + y⁸ z^(−7))</strong>. Applying the distributive property to this expression yields <strong>(x^(−6) y³ z⁵) (x⁴ z⁵) + (x^(−6) y³ z⁵) (y⁸ z^(−7))</strong>, or <strong>x^(−6) x⁴ y³ z⁵ z^(5 + x^(−6)) y³ y⁸ z⁵ z^(−7)</strong>. This expression is equivalent to <strong>x^(−6 + 4) y³ z^(5 + 5 + x^(−6)) y^(3 + 8) z^(5 − 7)</strong>, or <strong>x^(−2) y³ z^(10 + x^(−6)) y¹¹ z^(−2)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "dd4ab4c4",
    type: "mcq",
    questionHtml: "Which of the following is a factor of the polynomial above?",
    choices: [
      { label: "A", text: "<strong>a + b</strong>" },
      { label: "B", text: "<strong>2 a + 5 b</strong>" },
      { label: "C", text: "<strong>4 a + 5 b</strong>" },
      { label: "D", text: "<strong>4 a + 25 b</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The first and last terms of the polynomial are both squares such that <strong>4 a, ² = (2 a, ), ²</strong> and <strong>25 b² = (5 b, ), ²</strong>. The second term is twice the product of the square root of the first and last terms: <strong>20 a, b = 2 · 2 a · 5 b</strong>. Therefore, the polynomial is the square of a binomial such that <strong>4 a, ² + 20 a, b + 25 b² = (2 a + 5 b, ), ²</strong>, and <strong>(2 a + 5 b, )</strong> is a factor.Choice A is incorrect and may be the result of incorrectly factoring the polynomial. Choice C is incorrect and may be the result of dividing the second and third terms of the polynomial by their greatest common factor. Choice D is incorrect and may be the result of not factoring the coefficients.",
    hasFigure: false,
  },
  {
    id: "f237ccfc",
    type: "spr",
    questionHtml:
      "The sum of <strong>−2 x² + x + 31</strong> and <strong>3 x² + 7 x − 8</strong> can be written in the form <strong>a, x² + b x + c</strong>, where a, b, and c are constants. What is the value of <strong>a + b + c</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 32. The sum of the given expressions is <strong>(−2, x² + x + 31, ) + (3 x² + 7 x − 8, )</strong>. Combining like terms yields <strong>x² + 8 x + 23</strong>. Based on the form of the given equation, <strong>a = 1</strong>, <strong>b = 8</strong>, and <strong>c = 23</strong>. Therefore, <strong>a + b + c = 32</strong>.Alternate approach: Because <strong>a + b + c</strong> is the value of <strong>a, x² + b x + c</strong> when <strong>x = 1</strong>, it is possible to first make that substitution into each polynomial before adding them. When <strong>x = 1</strong>, the first polynomial is equal to<strong>−2 + 1 + 31 = 30</strong> and the second polynomial is equal to <strong>3 + 7 − 8 = 2</strong>. The sum of 30 and 2 is 32.",
    hasFigure: false,
  },
];

export const EQUIV_EXPR_HARD: PracticeQuestion[] = [
  {
    id: "12e7faf8",
    type: "mcq",
    questionHtml:
      "The equation <strong>the fraction with numerator x² + 6 x − 7, and denominator x + 7, end fraction = a x + d</strong> is true for all <strong>x ≠ −7</strong>, where a and d are integers. What is the value of <strong>a + d</strong> ?",
    choices: [
      { label: "A", text: "<strong>−6</strong>" },
      { label: "B", text: "<strong>−1</strong>" },
      { label: "C", text: "<strong>0</strong>" },
      { label: "D", text: "<strong>1</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since the expression <strong>x² + 6 x − 7</strong> can be factored as <strong>(x + 7, ) · (x − 1, )</strong>, the given equation can be rewritten as <strong>the fraction with numerator (x + 7, ) · (x − 1, ), and denominator x + 7, end fraction = a, x + d</strong>. Since <strong>x ≠ −7</strong>, <strong>x + 7</strong> is also not equal to 0, so both the numerator and denominator of <strong>the fraction with numerator (x + 7, ) · (x − 1, ), and denominator x + 7, end fraction</strong> can be divided by <strong>x + 7</strong>. This gives <strong>x − 1 = a x + d</strong>. Equating the coefficient of x on each side of the equation gives <strong>a = 1</strong>. Equating the constant terms gives <strong>d = −1</strong>. The sum is <strong>1 + −1 = 0</strong>.Choice A is incorrect and may result from incorrectly simplifying the equation. Choices B and D are incorrect. They are the values of d and a, respectively, not <strong>a + d</strong>.",
    hasFigure: false,
  },
  {
    id: "137cc6fd",
    type: "spr",
    questionHtml:
      "<strong>RootIndex 5 √(70 n) (RootIndex 6 √(70 n))²</strong><br>For what value of <strong>x</strong> is the given expression equivalent to <strong>(70 n)^(30 x)</strong>, where <strong>n > 1</strong>?",
    choices: [],
    correctAnswer: ".0177, .0178, 4/225",
    explanation:
      "The correct answer is <strong>(4) / (225)</strong>. An expression of the form <strong>RootIndex k √(a)</strong>, where <strong>k</strong> is an integer greater than <strong>1</strong> and <strong>a > or = 0</strong>, is equivalent to <strong>a^((1) / (k))</strong>. Therefore, the given expression, where <strong>n > 1</strong>, is equivalent to <strong>(70 n)^(one fifth) ((70 n)^(one sixth))²</strong>. Applying properties of exponents, this expression can be rewritten as <strong>(70 n)^(one fifth) (70 n)^(one sixth dot 2)</strong>, or <strong>(70 n)^(one fifth) (70 n)^(one third)</strong>, which can be rewritten as <strong>(70 n)^(one fifth + one third)</strong>, or <strong>(70 n)^(eight fifteenths)</strong>. It's given that the expression <strong>RootIndex 5 √(70 n) (RootIndex 6 √(70 n))²</strong> is equivalent to <strong>(70 n)^(30 x)</strong>, where <strong>n > 1</strong>. It follows that <strong>(70 n)^(eight fifteenths)</strong> is equivalent to <strong>(70 n)^(30 x)</strong>. Therefore, <strong>eight fifteenths = 30 x</strong>. Dividing both sides of this equation by <strong>30</strong> yields <strong>(8) / (450) = x</strong>, or <strong>(4) / (225) = x</strong>. Thus, the value of <strong>x</strong> for which the given expression is equivalent to <strong>(70 n)^(30 x)</strong>, where <strong>n > 1</strong>, is <strong>(4) / (225)</strong>. Note that 4/225, .0177, .0178, 0.017, and 0.018 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "20291f47",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(y + 12) / (x − 8) + (y (x − 8)) / (x² y − 8 xy)</strong>?",
    choices: [
      {
        label: "A",
        text: "<strong>(xy + y + 4) / (x³ y − 16 x² y + 64 xy)</strong>",
      },
      {
        label: "B",
        text: "<strong>(xy + 9 y + 12) / (x² y − 8 xy + x − 8)</strong>",
      },
      {
        label: "C",
        text: "<strong>(xy² + 13 xy − 8 y) / (x² y − 8 xy)</strong>",
      },
      {
        label: "D",
        text: "<strong>(xy² + 13 xy − 8 y) / (x³ y − 16 x² y + 64 xy)</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Factoring the denominator in the second term of the given expression gives <strong>(y + 12) / (x − 8) + (y (x − 8)) / (xy (x − 8))</strong>. This expression can be rewritten with common denominators by multiplying the first term by <strong>(xy) / (xy)</strong>, giving <strong>(xy (y + 12)) / (xy (x − 8)) + (y (x − 8)) / (xy (x − 8))</strong>. Adding these two terms yields <strong>(xy (y + 12) + y (x − 8)) / (xy (x − 8))</strong>. Using the distributive property to rewrite this expression gives <strong>(xy² + 12 xy + xy − 8 y) / (x² y − 8 xy)</strong>. Combining the like terms in the numerator of this expression gives <strong>(xy² + 13 xy − 8 y) / (x² y − 8 xy)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "22fd3e1f",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to <strong>the fraction f of x over g of x</strong>, for <strong>x > 3</strong> ?",
    choices: [
      {
        label: "A",
        text: "<strong>the fraction with numerator 1, and denominator x + 1, end fraction</strong>",
      },
      {
        label: "B",
        text: "<strong>the fraction with numerator x + 3, and denominator x + 1, end fraction</strong>",
      },
      {
        label: "C",
        text: "<strong>the fraction with numerator x · (x − 3, ), and denominator x + 1, end fraction</strong>",
      },
      {
        label: "D",
        text: "<strong>the fraction with numerator x · (x + 3, ), and denominator x + 1, end fraction</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since <strong>x³ − 9 x = x · (x + 3, ) · (x − 3, )</strong> and <strong>x² − 2 x − 3 = (x + 1, ) · (x − 3, )</strong>, the fraction<strong>f of x, over g of x</strong> can be written as <strong>the fraction with numerator x · (x + 3, ) · (x − 3, ), and denominator, (x + 1, ) · (x − 3, ), end fraction</strong>. It is given that <strong>x > 3</strong>, so the common factor <strong>x − 3</strong> is not equal to 0. Therefore, the fraction can be further simplified to <strong>the fraction with numerator x · (x + 3, ), and denominator x + 1, end fraction</strong>.Choice A is incorrect. The expression <strong>the fraction 1 over, x + 1, end fraction</strong> is not equivalent to <strong>the fraction f of x, over g of x</strong> because at <strong>x = 0</strong>, <strong>the fraction 1 over, x + 1, end fraction</strong> as a value of 1 and <strong>the fraction f of x, over g of x</strong> has a value of 0.<br>Choice B is incorrect and results from omitting the factor x in the factorization of <strong>f of x</strong>. Choice C is incorrect and may result from incorrectly factoring <strong>g of x</strong> as <strong>(x + 1, ) · (x + 3, )</strong> instead of <strong>(x + 1, ) · (x − 3, )</strong>.",
    hasFigure: false,
  },
  {
    id: "2c88af4d",
    type: "mcq",
    questionHtml:
      "The expression <strong>the fraction with numerator x to the power − 2, end power · y to the power one-half, and denominator x to the power one-third, end power · y to the power − 1, end fraction</strong>, where <strong>x > 1</strong> and <strong>y > 1</strong>, is equivalent to which of the following?",
    choices: [
      {
        label: "A",
        text: "<strong>The fraction with numerator the √ y and denominator the cube root of x²</strong>",
      },
      {
        label: "B",
        text: "<strong>The fraction with numerator y · the √ y and denominator the cube root of x²</strong>",
      },
      {
        label: "C",
        text: "<strong>The fraction with numerator y · the √ y and denominator x · the √ x</strong>",
      },
      {
        label: "D",
        text: "<strong>The fraction with numerator y · the √ y and denominator x² · the cube root of x</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. For <strong>x > 1</strong> and <strong>y > 1</strong>, <strong>x to the one third power</strong> and <strong>y to the one half power</strong> are equivalent to <strong>the cube root of x</strong> and <strong>the √ y</strong>, respectively. Also, <strong>x to the − 2 power</strong> and <strong>y to the − 1 power</strong> are equivalent to <strong>the fraction 1 over x², end fraction</strong> and <strong>1 over y</strong>, respectively. Therefore, the given expression can be rewritten as <strong>the fraction with numerator, y · the √ y, and denominator x² · the cube root of x, end fraction</strong>.Choices A, B, and C are incorrect because these choices are not equivalent to the given expression for <strong>x > 1</strong> and <strong>y > 1</strong>.<br>For example, for <strong>x = 2</strong> and <strong>y = 2</strong>, the value of the given expression is <strong>2 to the − five sixths power</strong>; the values of the choices, however, are <strong>2 to the − one third power</strong>, <strong>2 to the five sixths power</strong>, and 1, respectively.",
    hasFigure: false,
  },
  {
    id: "34847f8a",
    type: "mcq",
    questionHtml:
      "The equation above is true for all <strong>x > 2</strong>, where r and t are positive constants. What is the value of rt ?",
    choices: [
      { label: "A", text: "<strong>−20</strong>" },
      { label: "B", text: "<strong>15</strong>" },
      { label: "C", text: "<strong>20</strong>" },
      { label: "D", text: "<strong>60</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. To express the sum of the two rational expressions on the left-hand side of the equation as the single rational expression on the right-hand side of the equation, the expressions on the left-hand side must have the same denominator. Multiplying the first expression by <strong>the fraction with numerator x + 5, and denominator x − 5, end fraction</strong> results in <strong>the fraction with numerator 2 · (x + 5, ), and denominator, (x − 2, ) · (x + 5, ), end fraction</strong>, and multiplying the second expression by <strong>the fraction with numerator x − 2, and denominator x − 2, end fraction</strong> results in <strong>the fraction with numerator 3 · (x − 2, ), and denominator, (x − 2, ) · (x + 5, ), end fraction</strong>, so the given equation can be rewritten as <strong>the fraction with numerator 2 · (x + 5, ), and denominator, (x − 2, ) · (x + 5, ), end fraction + the fraction with numerator 3 · (x − 2, ), and denominator, (x − 2, ) · (x + 5, ), end fraction = the fraction with numerator r x + t, and denominator, (x − 2, ) · (x + 5, ), end fraction</strong>, or <strong>the fraction with numerator 2 x + 10, and denominator, (x − 2, ) · (x + 5, ), end fraction + the fraction with numerator 3 x − 6, and denominator, (x − 2, ) · (x + 5, ), end fraction = the fraction with numerator r x + t, and denominator, (x − 2, ) · (x + 5, ), end fraction</strong>. Since the two rational expressions on the left-hand side of the equation have the same denominator as the rational expression on the right-hand side of the equation, it follows that <strong>(2 x + 10, ) + (3 x − 6, ) = r x + t</strong>. Combining like terms on the left-hand side yields <strong>5 x + 4 = r x + t</strong>, so it follows that <strong>r = 5</strong> and <strong>t = 4</strong>. Therefore, the value of <strong>r t</strong> is <strong>5 · 4, which = 20</strong>.Choice A is incorrect and may result from an error when determining the sign of either r or t. Choice B is incorrect and may result from not distributing the 2 and 3 to their respective terms in <strong>the fraction with numerator 2 · (x + 5, ), and denominator, (x − 2, ) · (x + 5, ), end fraction + the fraction with numerator 3 · (x − 2, ), and denominator, (x − 2, ) · (x + 5, ), end fraction = the fraction with numerator r x + t, and denominator, (x − 2, ) · (x + 5, ), end fraction</strong>. Choice D is incorrect and may result from a calculation error.",
    hasFigure: false,
  },
  {
    id: "371cbf6b",
    type: "mcq",
    questionHtml:
      "The equation above is true for all x, where a and b are constants. What is the value of ab ?",
    choices: [
      { label: "A", text: "18" },
      { label: "B", text: "20" },
      { label: "C", text: "24" },
      { label: "D", text: "40" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent. Multiplying the polynomials on the left-hand side of the equation gives <strong>5 a, x³ − a, b x² + 4 a, x + 15 x² − 3 b x + 12</strong>. On the right-hand side of the equation, the only <strong>x²</strong>-term is <strong>−9 x²</strong>. Since the expressions on both sides of the equation are equivalent, it follows that <strong>−a, b x² + 15 x² = −9 x²</strong>, which can be rewritten as <strong>(−a, b + 15, ) · x² = −9 x²</strong>. Therefore, <strong>−a, b + 15 = −9</strong>, which gives <strong>a, b = 24</strong>.Choice A is incorrect. If <strong>a, b = 18</strong>, then the coefficient of <strong>x²</strong> on the left-hand side of the equation would be <strong>−18 + 15 = −3</strong>, which doesn’t equal the coefficient of <strong>x²</strong>, <strong>−9</strong>, on the right-hand side. Choice B is incorrect. If <strong>a, b = 20</strong>, then the coefficient of <strong>x²</strong> on the left-hand side of the equation would be <strong>−20 + 15 = −5</strong>, which doesn’t equal the coefficient of <strong>x²</strong>, <strong>−9</strong>, on the right-hand side. Choice D is incorrect. If <strong>a, b = 40</strong>, then the coefficient of <strong>x²</strong> on the left-hand side of the equation would be <strong>−40 + 15 = −25</strong>, which doesn’t equal the coefficient of <strong>x²</strong>, <strong>−9</strong>, on the right-hand side.",
    hasFigure: false,
  },
  {
    id: "40c09d66",
    type: "spr",
    questionHtml:
      "If <strong>the fraction with numerator the √ x to the fifth power, end root, and denominator the cube root of x to the fourth power, end root, end fraction = x raised to the fraction a over b power</strong> for all positive values of x, what is the value of <strong>the fraction a over b</strong>?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is <strong>7 over 6</strong>. The value of <strong>a, over b</strong> can be found by first rewriting the left-hand side of the given equation as <strong>the fraction with numerator x raised to the five halves power, and denominator x raised to the four thirds power, end fraction</strong>. Using the properties of exponents, this expression can be rewritten as <strong>x raised to the (five halves − four thirds, ), power</strong>. This expression can be rewritten by subtracting the fractions in the exponent, which yields <strong>x raised to the fraction 7 over 6, power</strong>. Thus, <strong>a, over b</strong> is <strong>7 over 6</strong>. Note that 7/6, 1.166, and 1.167 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "42f8e4b4",
    type: "spr",
    questionHtml:
      "One of the factors of <strong>2 x³ + 42 x² + 208 x</strong> is <strong>x + b</strong>, where <strong>b</strong> is a positive constant. What is the smallest possible value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "8",
    explanation:
      "The correct answer is <strong>8</strong>. Since each term of the given expression, <strong>2 x³ + 42 x² + 208 x</strong>, has a factor of <strong>2 x</strong>, the expression can be rewritten as <strong>2 x (x²) + 2 x (21 x) + 2 x (104)</strong>, or <strong>2 x (x² + 21 x + 104)</strong>. Since the values <strong>8</strong> and <strong>13</strong> have a sum of <strong>21</strong> and a product of <strong>104</strong>, the expression <strong>x² + 21 x + 104</strong> can be factored as <strong>(x + 8) (x + 13)</strong>. Therefore, the given expression can be factored as <strong>2 x (x + 8) (x + 13)</strong>. It follows that the factors of the given expression are <strong>2</strong>, <strong>x</strong>, <strong>x + 8</strong>, and <strong>x + 13</strong>. Of these factors, only <strong>x + 8</strong> and <strong>x + 13</strong> are of the form <strong>x + b</strong>, where <strong>b</strong> is a positive constant. Therefore, the possible values of <strong>b</strong> are <strong>8</strong> and <strong>13</strong>. Thus, the smallest possible value of <strong>b</strong> is <strong>8</strong>.",
    hasFigure: false,
  },
  {
    id: "433184f1",
    type: "mcq",
    questionHtml:
      "Which expression is equivalent to <strong>(4) / (4 x − 5) − (1) / (x + 1)</strong>?",
    choices: [
      { label: "A", text: "<strong>(1) / ((x + 1) (4 x − 5))</strong>" },
      { label: "B", text: "<strong>(3) / (3 x − 6)</strong>" },
      { label: "C", text: "<strong>− (1) / ((x + 1) (4 x − 5))</strong>" },
      { label: "D", text: "<strong>(9) / ((x + 1) (4 x − 5))</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The expression <strong>(4) / (4 x − 5) − (1) / (x + 1)</strong> can be rewritten as <strong>(4) / (4 x − 5) + ((−1)) / (x + 1)</strong>. To add the two terms of this expression, the terms can be rewritten with a common denominator. Since <strong>(x + 1) / (x + 1) = 1</strong>, the expression <strong>(4) / (4 x − 5)</strong> can be rewritten as <strong>((x + 1) (4)) / ((x + 1) (4 x − 5))</strong>. Since <strong>(4 x − 5) / (4 x − 5) = 1</strong>, the expression <strong>(−1) / (x + 1)</strong> can be rewritten as <strong>((4 x − 5) (−1)) / ((4 x − 5) (x + 1))</strong>. Therefore, the expression <strong>(4) / (4 x − 5) + ((−1)) / (x + 1)</strong> can be rewritten as <strong>((x + 1) (4)) / ((x + 1) (4 x − 5)) + ((4 x − 5) (−1)) / ((4 x − 5) (x + 1))</strong>, which is equivalent to <strong>((x + 1) (4) + (4 x − 5) (−1)) / ((x + 1) (4 x − 5))</strong>. Applying the distributive property to each term of the numerator yields <strong>((4 x + 4) + (−4 x + 5)) / ((x + 1) (4 x − 5))</strong>, or  <strong>((4 x + (−4 x)) + (4 + 5)) / ((x + 1) (4 x − 5))</strong>. Adding like terms in the numerator yields <strong>(9) / ((x + 1) (4 x − 5))</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "5355c0ef",
    type: "spr",
    questionHtml:
      "<strong>0.36 x² + 0.63 x + 1.17</strong><br>The given expression can be rewritten as <strong>a (4 x² + 7 x + 13)</strong>, where <strong>a</strong> is a constant. What is the value of <strong>a</strong>?",
    choices: [],
    correctAnswer: ".09, 9/100",
    explanation:
      "The correct answer is <strong>.09</strong>. It's given that the expression <strong>0.36 x² + 0.63 x + 1.17</strong> can be rewritten as <strong>a (4 x² + 7 x + 13)</strong>. Applying the distributive property to the expression <strong>a (4 x² + 7 x + 13)</strong> yields <strong>4 a x² + 7 a x + 13 a</strong>. Therefore, <strong>0.36 x² + 0.63 x + 1.17</strong> can be rewritten as <strong>4 a x² + 7 a x + 13 a</strong>. It follows that in the expressions <strong>0.36 x² + 0.63 x + 1.17</strong> and <strong>4 a x² + 7 a x + 13 a</strong>, the coefficients of <strong>x²</strong> are equivalent, the coefficients of <strong>x</strong> are equivalent, and the constant terms are equivalent. Therefore, <strong>0.36 = 4 a</strong>, <strong>0.63 = 7 a</strong>, and <strong>1.17 = 13 a</strong>. Solving any of these equations for <strong>a</strong> yields the value of <strong>a</strong>. Dividing both sides of the equation <strong>0.36 = 4 a</strong> by <strong>4</strong> yields <strong>0.09 = a</strong>. Therefore, the value of <strong>a</strong> is <strong>0.09</strong>. Note that .09 and 9/100 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "89fc23af",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is equivalent to <strong>the fraction with numerator x² − 2 x − 5, and denominator x − 3</strong> ?",
    choices: [
      {
        label: "A",
        text: "<strong>x − 5 − the fraction with numerator 20, and denominator x − 3</strong>",
      },
      {
        label: "B",
        text: "<strong>x − 5 − the fraction with numerator 10, and denominator x − 3</strong>",
      },
      {
        label: "C",
        text: "<strong>x + 1 − the fraction with numerator 8, and denominator x − 3</strong>",
      },
      {
        label: "D",
        text: "<strong>x + 1 − the fraction with numerator 2, and denominator x − 3</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The numerator of the given expression can be rewritten in terms of the denominator, <strong>x − 3</strong>, as follows: <strong>x² − 2 x − 5 = x² − 3 x + x − 3 − 2</strong>, which is equivalent to <strong>x · (x − 3, ) + (x − 3, ) − 2</strong>. So the given expression is equivalent to <strong>the fraction with numerator x · (x − 3, ) + (x − 3, ) − 2, and denominator x − 3, end fraction = the fraction with numerator x · (x − 3, ), and denominator x − 3, end fraction + the fraction with numerator x − 3, and denominator x − 3, end fraction − the fraction with numerator 2, and denominator x − 3, end fraction</strong>. Since the given expression is defined for <strong>x ≠ 3</strong>, the expression can be rewritten as <strong>x + 1 − the fraction with numerator 2, and denominator x − 3, end fraction</strong>.Long division can also be used as an alternate approach. Choices A, B, and C are incorrect and may result from errors made when dividing the two polynomials or making use of structure.",
    hasFigure: false,
  },
  {
    id: "911c415b",
    type: "spr",
    questionHtml:
      "The expression above can be written in the form <strong>a · y² + b</strong>, where a and b are constants. What is the value of <strong>a + b</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 6632. Applying the distributive property to the expression yields <strong>(7532 + 100 y², ) + (100 y² − 1100, )</strong>. Then adding together <strong>7532 + 100 y²</strong> and <strong>100 y² − 1100</strong> and collecting like terms results in <strong>200 y² + 6432</strong>. This is written in the form <strong>a, y² + b</strong>, where <strong>a = 200</strong> and <strong>b = 6432</strong>. Therefore <strong>a + b = 200 + 6432, which = 6632</strong>.",
    hasFigure: false,
  },
  {
    id: "a0b4103e",
    type: "mcq",
    questionHtml:
      "The expression <strong>one-third x² − 2</strong> can be rewritten as <strong>one-third · (x − k, ) · (x + k, )</strong>, where k is a positive constant. What is the value of k ?",
    choices: [
      { label: "A", text: "2" },
      { label: "B", text: "6" },
      { label: "C", text: "<strong>the √ 2</strong>" },
      { label: "D", text: "<strong>the √ 6</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Factoring out the coefficient <strong>one third</strong>, the given expression can be rewritten as <strong>one third · (x² − 6, )</strong>. The expression <strong>x² − 6</strong> can be approached as a difference of squares and rewritten as <strong>(x − the √ 6, ) · (x + the √ 6, )</strong>. Therefore, k must be <strong>the √ 6</strong>.Choice A is incorrect. If k were 2, then the expression given would be rewritten as <strong>one third · (x − 2, ) · (x + 2, )</strong>, which is equivalent to <strong>one third, x² − four thirds</strong>, not <strong>one third, x² − 2</strong>.<br>Choice B is incorrect. This may result from incorrectly factoring the expression and finding <strong>(x − 6, ) · (x + 6, )</strong> as the factored form of the expression. Choice C is incorrect. This may result from incorrectly distributing the <strong>one third</strong> and rewriting the expression as <strong>one third · (x² − 2, )</strong>.",
    hasFigure: false,
  },
  {
    id: "ad038c19",
    type: "mcq",
    questionHtml:
      "Which of the following is equivalent to <strong>parenthesis, a + the fraction b over 2, ), ²</strong> ?",
    choices: [
      { label: "A", text: "<strong>a, ² + the fraction b² over 2</strong>" },
      { label: "B", text: "<strong>a, ² + the fraction b² over 4</strong>" },
      {
        label: "C",
        text: "<strong>a, ² + the fraction a · b, over 2, end fraction + the fraction b² over 2</strong>",
      },
      {
        label: "D",
        text: "<strong>a, ² + a · b + the fraction b² over 4</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The expression <strong>(a + the fraction b over 2, ), ²</strong> can be rewritten as <strong>(a + the fraction b over 2, ) · (a + the fraction b over 2, )</strong>. Using the distributive property, the expression yields <strong>(a + the fraction b over 2, ) · (a + the fraction b over 2, ) = a, ² + the fraction with numerator a, b and denominator 2, end fraction + the fraction with numerator a, b and denominator 2, end fraction + the fraction b², over 4</strong>. Combining like terms gives <strong>a, ² + a, b + the fraction b² over 4</strong>.Choices A, B, and C are incorrect and may result from errors using the distributive property on the given expression or combining like terms.",
    hasFigure: false,
  },
  {
    id: "b74f2feb",
    type: "spr",
    questionHtml:
      "The expression <strong>6 RootIndex 5 √(3⁵ x⁴⁵) dot RootIndex 8 √(2⁸ x)</strong> is equivalent to <strong>a x^(b)</strong>, where <strong>a</strong> and <strong>b</strong> are positive constants and <strong>x > 1</strong>. What is the value of <strong>a + b</strong>?",
    choices: [],
    correctAnswer: "361/8, 45.12, 45.13",
    explanation:
      "The correct answer is <strong>(361) / (8)</strong>. The rational exponent property is <strong>RootIndex n √(y^(m)) = y^((m) / (n))</strong>, where <strong>y > 0</strong>, <strong>m</strong> and <strong>n</strong> are integers, and <strong>n > 0</strong>. This property can be applied to rewrite the given expression <strong>6 RootIndex 5 √(3⁵ x⁴⁵) dot RootIndex 8 √(2⁸ x)</strong> as <strong>6 (3^(five fifths)) (x^((45) / (5))) (2^(eight eighths)) (x^(one eighth))</strong>, or <strong>6 (3) (x⁹) (2) (x^(one eighth))</strong>. This expression can be rewritten by multiplying the constants, which gives <strong>36 (x⁹) (x^(one eighth))</strong>. The multiplication exponent property is <strong>y^(n) dot y^(m) = y^(n + m)</strong>, where <strong>y > 0</strong>. This property can be applied to rewrite the expression <strong>36 (x⁹) (x^(one eighth))</strong> as <strong>36 x^(9 + one eighth)</strong>, or <strong>36 x^((73) / (8))</strong>. Therefore, <strong>6 RootIndex 5 √(3⁵ x⁴⁵) dot RootIndex 8 √(2⁸ x) = 36 x^((73) / (8))</strong>. It's given that <strong>6 RootIndex 5 √(3⁵ x⁴⁵) dot RootIndex 8 √(2⁸ x)</strong> is equivalent to <strong>a x^(b)</strong>; therefore, <strong>a = 36</strong> and <strong>b = (73) / (8)</strong>. It follows that <strong>a + b = 36 + (73) / (8)</strong>. Finding a common denominator on the right-hand side of this equation gives <strong>a + b = (288) / (8) + (73) / (8)</strong>, or <strong>a + b = (361) / (8)</strong>. Note that 361/8, 45.12, and 45.13 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "c3b116d7",
    type: "mcq",
    questionHtml:
      "Which of the following expressions is(are) a factor of <strong>3 x² + 20 x − 63</strong>?<br><br> <strong>x − 9</strong><br><strong>3 x − 7</strong>",
    choices: [
      { label: "A", text: "I only" },
      { label: "B", text: "II only" },
      { label: "C", text: "I and II" },
      { label: "D", text: "Neither I nor II" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The given expression can be factored by first finding two values whose sum is <strong>20</strong> and whose product is <strong>3 (−63)</strong>, or <strong>−189</strong>. Those two values are <strong>27</strong> and <strong>−7</strong>. It follows that the given expression can be rewritten as <strong>3 x² + 27 x − 7 x − 63</strong>. Since the first two terms of this expression have a common factor of <strong>3 x</strong> and the last two terms of this expression have a common factor of <strong>−7</strong>, this expression can be rewritten as <strong>3 x (x + 9) − 7 (x + 9)</strong>. Since the two terms of this expression have a common factor of <strong>(x + 9)</strong>, it can be rewritten as <strong>(3 x − 7) (x + 9)</strong>. Therefore, expression II, <strong>3 x − 7</strong>, is a factor of <strong>3 x² + 20 x − 63</strong>, but expression I, <strong>x − 9</strong>, is not a factor of <strong>3 x² + 20 x − 63</strong>. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "c81b6c57",
    type: "mcq",
    questionHtml:
      "In the expression [] , p is a constant. This expression is equivalent to the expression <strong>6 x² − 155 x + 24</strong>. What is the value of p ?",
    choices: [
      { label: "A", text: "<strong>−3</strong>" },
      { label: "B", text: "<strong>7</strong>" },
      { label: "C", text: "<strong>13</strong>" },
      { label: "D", text: "<strong>155</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Using the distributive property, the first given expression can be rewritten as 6x2 + 3px + 24 – 16px – 64x + 24, and then rewritten as 6x2 + (3p – 16p – 64)x + 24. Since the expression 6x2 + (3p – 16p – 64)x + 24 is equivalent to 6x2 – 155x + 24, the coefficients of the x terms from each expression are equivalent to each other; thus 3p – 16p – 64 = –155. Combining like terms gives –13p  – 64 = –155. Adding 64 to both sides of the equation gives –13p = –71. Dividing both sides of the equation by –13 yields p = 7.Choice A is incorrect. If p = –3, then the first expression would be equivalent to 6x2 – 25x + 24. Choice C is incorrect. If p = 13, then the first expression would be equivalent to 6x2 – 233x + 24. Choice D is incorrect. If p = 155, then the first expression would be equivalent to 6x2 – 2,079x + 24.",
    hasFigure: false,
  },
  {
    id: "d8789a4c",
    type: "mcq",
    questionHtml:
      "In the expression above, b and c are positive integers. If the expression is equivalent to <strong>x + b</strong> and <strong>x ≠ b</strong>, which of the following could be the value of c ?",
    choices: [
      { label: "A", text: "4" },
      { label: "B", text: "6" },
      { label: "C", text: "8" },
      { label: "D", text: "10" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. If the given expression is equivalent to <strong>x + b</strong>, then <strong>the fraction with numerator x² − c, and denominator x − b, end fraction = x + b</strong>, where x isn’t equal to b. Multiplying both sides of this equation by <strong>x − b</strong> yields <strong>x² − c = (x + b, ) · (x − b, )</strong>. Since the right-hand side of this equation is in factored form for the difference of squares, the value of c must be a perfect square. Only choice A gives a perfect square for the value of c.Choices B, C, and D are incorrect. None of these values of c produces a difference of squares. For example, when 6 is substituted for c in the given expression, the result is <strong>the fraction with numerator x² − 6, and denominator x − b, end fraction</strong>. The expression <strong>x² − 6</strong> can’t be factored with integer values, and therefore <strong>the fraction with numerator x² − 6, and denominator x − b, end fraction</strong> isn’t equivalent to <strong>x + b</strong>.",
    hasFigure: false,
  },
  {
    id: "e117d3b8",
    type: "mcq",
    questionHtml:
      "If a and c are positive numbers, which of the following is equivalent to <strong>the √, (a + c, ), ³, end root · the √ a + c, end root</strong>?",
    choices: [
      { label: "A", text: "<strong>a + c</strong>" },
      { label: "B", text: "<strong>a² + c²</strong>" },
      { label: "C", text: "<strong>a² + 2 a c + c²</strong>" },
      { label: "D", text: "<strong>a² · c²</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Using the property that <strong>the √ x, end root · the √ y, end root = the √ xy, end root</strong> for positive numbers x and y, with x = (a + c)3 and y = a + c, it follows that <strong>the √, (a + c, ), ³, end root · the √ a + c, end root = the √, (a + c, end parenthesis to the power 4, end root</strong>. By rewriting (a + c)4 as ((a + c)2)2, it is possible to simplify the square root expression as follows: <strong>the √, open outer parenthesis, open inner parenthesis, a + c, close inner parenthesis, ², close outer parenthesis, ², end root = (a + c, ), ², which = a, ² + 2 a, c + c²</strong>.<br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> <br> Choice A is incorrect and may be the result of <strong>the √, (a + c, ), ³, end root ÷ the √, a + c, end root</strong>. Choice B is incorrect and may be the result of incorrectly rewriting (a + c)2 as a2 + c2. Choice D is incorrect and may be the result of incorrectly applying properties of exponents.",
    hasFigure: false,
  },
  {
    id: "ea6d05bb",
    type: "spr",
    questionHtml:
      "The expression <strong>(3 x − 23) (19 x + 6)</strong> is equivalent to the expression <strong>a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants. What is the value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "-419",
    explanation:
      "The correct answer is <strong>−419</strong>. It's given that the expression <strong>(3 x − 23) (19 x + 6)</strong> is equivalent to the expression <strong>a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants. Applying the distributive property to the given expression, <strong>(3 x − 23) (19 x + 6)</strong>, yields <strong>(3 x) (19 x) + (3 x) (6) − (23) (19 x) − (23) (6)</strong>, which can be rewritten as <strong>57 x² + 18 x − 437 x − 138</strong>. Combining like terms yields <strong>57 x² − 419 x − 138</strong>. Since this expression is equivalent to <strong>a x² + b x + c</strong>, it follows that the value of <strong>b</strong> is <strong>−419</strong>.",
    hasFigure: false,
  },
  {
    id: "f89e1d6f",
    type: "mcq",
    questionHtml:
      "If <strong>a = c + d</strong>, which of the following is equivalent to the expression <strong>x² − c² − 2 c d − d²</strong>?",
    choices: [
      { label: "A", text: "<strong>(x + a, ), ²</strong>" },
      { label: "B", text: "<strong>(x − a, ), ²</strong>" },
      { label: "C", text: "<strong>(x + a, ) · (x − a, )</strong>" },
      { label: "D", text: "<strong>x² − a · x − a²</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Factoring –1 from the second, third, and fourth terms gives x2 – c2 – 2cd – d2 = x2 – (c2 + 2cd + d2). The expression c2 + 2cd + d2 is the expanded form of a perfect square: c2 + 2cd + d2 = (c + d)2. Therefore, x2 – (c2 + 2cd + d2) = x2 – (c + d)2. Since a = c + d, x2 – (c + d)2 = x2 – a2. Finally, because x2 – a2 is the difference of squares, it can be expanded as x2 – a2 = (x + a)(x – a).Choices A and B are incorrect and may be the result of making an error in factoring the difference of squares x2 – a2. Choice D is incorrect and may be the result of incorrectly combining terms.",
    hasFigure: false,
  },
  {
    id: "ffdbcad4",
    type: "mcq",
    questionHtml:
      "The expression <strong>4 x² + b x − 45</strong>, where <strong>b</strong> is a constant, can be rewritten as <strong>(h x + k) (x + j)</strong>, where <strong>h</strong>, <strong>k</strong>, and <strong>j</strong> are integer constants. Which of the following must be an integer?",
    choices: [
      { label: "A", text: "<strong>(b) / (h)</strong>" },
      { label: "B", text: "<strong>(b) / (k)</strong>" },
      { label: "C", text: "<strong>(45) / (h)</strong>" },
      { label: "D", text: "<strong>(45) / (k)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that <strong>4 x² + b x − 45</strong> can be rewritten as <strong>(h x + k) (x + j)</strong>. The expression <strong>(h x + k) (x + j)</strong> can be rewritten as <strong>h x² + j h x + k x + k j</strong>, or <strong>h x² + (j h + k) x + k j</strong>. Therefore, <strong>h x² + (j h + k) x + k j</strong> is equivalent to <strong>4 x² + b x − 45</strong>. It follows that <strong>k j = −45</strong>. Dividing each side of this equation by <strong>k</strong> yields <strong>j = (−45) / (k)</strong>. Since <strong>j</strong> is an integer, <strong>− (45) / (k)</strong> must be an integer. Therefore, <strong>(45) / (k)</strong> must also be an integer.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];
