GRE Quiz: Quadratic Polynomial Relationships
17 questions · exam conditions
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Quadratic Polynomial RelationshipsQuestion 1 of 17

If h(x)=x2+2x−15h(x)=x^2+2x-15, which of the following is a factor of h(x)h(x)?

x+3x+3
x−3x-3
x+5x+5
x−15x-15
x+15x+15
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GRE Quiz: Quadratic Polynomial Relationships

Practice Quadratic Polynomial Relationships in GRE with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Quadratic Polynomial Relationships, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

If h(x)=x2+2x−15h(x)=x^2+2x-15, which of the following is a factor of h(x)h(x)?

  1. x+3x+3
  2. x−3x-3
  3. x+5x+5 (correct answer)
  4. x−15x-15
  5. x+15x+15

Explanation: This question tests finding factors of a quadratic polynomial. To find factors of h(x) = x² + 2x - 15, we need to factor the expression. We look for two numbers that multiply to -15 and add to 2; these are 5 and -3. Therefore, h(x) = (x + 5)(x - 3), giving us factors (x + 5) and (x - 3). Choice C correctly identifies (x + 5) as a factor. A common mistake is to confuse the signs in factoring, potentially selecting (x - 5) or mixing up which values work.

Question 2

If xx is a real number and x2−5x+6=0x^2-5x+6=0, which of the following is a possible value of xx?

  1. −2-2
  2. 22 (correct answer)
  3. −3-3
  4. 66
  5. 32\frac{3}{2}

Explanation: This question tests quadratic polynomial relationships by asking us to solve a quadratic equation. To find possible values of x, we need to factor the quadratic expression x² - 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5; these are -2 and -3. Therefore, x² - 5x + 6 = (x - 2)(x - 3) = 0, giving us x = 2 or x = 3. The correct answer is B, which gives x = 2 as a possible value. A common error would be to confuse the signs when factoring, leading to incorrect values like x = -2.

Question 3

Which of the following is equivalent to the polynomial expression x2−9x^2-9?

  1. (x−9)(x+1)(x-9)(x+1)
  2. (x−3)2(x-3)^2
  3. (x−3)(x+3)(x-3)(x+3) (correct answer)
  4. (x+3)2(x+3)^2
  5. x(x−9)x(x-9)

Explanation: This question tests polynomial factorization, specifically recognizing a difference of squares pattern. The expression x² - 9 follows the pattern a² - b² = (a + b)(a - b), where a = x and b = 3. Applying this formula, we get x² - 9 = x² - 3² = (x + 3)(x - 3). Therefore, the correct factorization is (x - 3)(x + 3), which matches choice C. A common mistake is to factor this as (x - 3)², which would expand to x² - 6x + 9, not x² - 9.

Question 4

Which of the following statements about the quadratic function g(x)=−2x2+8x+3g(x)=-2x^2+8x+3 is true?

  1. The parabola opens upward because the coefficient of x2x^2 is negative.
  2. The parabola opens downward because the coefficient of x2x^2 is negative. (correct answer)
  3. The parabola has a yy-intercept of 88.
  4. The parabola has no real xx-intercepts.
  5. The vertex occurs at x=8x=8.

Explanation: This question tests understanding of quadratic function properties, particularly the effect of the leading coefficient. In the function g(x) = -2x² + 8x + 3, the coefficient of x² is -2, which is negative. When the leading coefficient of a quadratic is negative, the parabola opens downward, making choice B correct. The y-intercept occurs when x = 0, giving g(0) = 3, not 8 as choice C suggests. Choice A incorrectly states that the parabola opens upward despite acknowledging the negative coefficient.

Question 5

For the polynomial f(x)=x2−4x−5f(x)=x^2-4x-5, what is the value of f(5)−f(−1)f(5)-f(-1)?

  1. 00 (correct answer)
  2. −20-20
  3. 2020
  4. −10-10
  5. 1010

Explanation: This question tests evaluating polynomial functions and computing differences. To find f(5) - f(-1), we must evaluate f(x) = x² - 4x - 5 at both x = 5 and x = -1. For f(5): f(5) = 5² - 4(5) - 5 = 25 - 20 - 5 = 0. For f(-1): f(-1) = (-1)² - 4(-1) - 5 = 1 + 4 - 5 = 0. Therefore, f(5) - f(-1) = 0 - 0 = 0, giving us answer A. A common error would be to make arithmetic mistakes when substituting negative values, potentially leading to incorrect answers like -20 or 20.

Question 6

A polynomial is defined by p(x)=(x−1)(x+4)(x−2)p(x)=(x-1)(x+4)(x-2). Which of the following statements is true?

  1. p(x)p(x) has zeros at x=−1,4,2x=-1,4,2.
  2. p(x)p(x) has exactly two real zeros.
  3. p(x)p(x) has a zero at x=0x=0.
  4. p(x)p(x) has zeros at x=1,−4,2x=1,-4,2. (correct answer)
  5. p(x)p(x) has a double zero at x=1x=1.

Explanation: This question tests understanding of polynomial zeros from factored form. When a polynomial is written as p(x) = (x - 1)(x + 4)(x - 2), the zeros occur when any factor equals zero. Setting each factor to zero: x - 1 = 0 gives x = 1; x + 4 = 0 gives x = -4; x - 2 = 0 gives x = 2. Therefore, the polynomial has zeros at x = 1, -4, and 2, which matches choice D. Choice A incorrectly identifies the zero from (x + 4) as x = 4 instead of x = -4, a common sign error when working with factored forms.

Question 7

The height (in meters) of a ball tt seconds after it is thrown is given by H(t)=−5t2+20t+1H(t)=-5t^2+20t+1. At what time tt does the ball reach its maximum height?

  1. 11
  2. 22 (correct answer)
  3. 44
  4. 55
  5. 1010

Explanation: This question tests finding the maximum of a quadratic function in a real-world context. For a quadratic function in the form H(t) = at² + bt + c with a < 0, the maximum occurs at t = -b/(2a). In H(t) = -5t² + 20t + 1, we have a = -5 and b = 20. Therefore, the maximum occurs at t = -20/(2(-5)) = -20/(-10) = 2 seconds. The answer is B. A common error would be to use the wrong formula or make sign errors in the calculation, potentially getting t = 4 or another incorrect value.

Question 8

If q(x)=x3−3x2−4x+12q(x)=x^3-3x^2-4x+12, which of the following is the value of q(3)q(3)?

  1. 1212
  2. 66
  3. −6-6
  4. 00 (correct answer)
  5. −12-12

Explanation: This question tests evaluating a cubic polynomial at a specific value. To find q(3) where q(x) = x³ - 3x² - 4x + 12, we substitute x = 3 into the expression. Computing step by step: q(3) = 3³ - 3(3²) - 4(3) + 12 = 27 - 3(9) - 12 + 12 = 27 - 27 - 12 + 12 = 0. Therefore, q(3) = 0, making A the correct answer. A common error would be arithmetic mistakes in computing powers or handling negative terms, potentially leading to answers like 6 or -6.

Question 9

For the quadratic function f(x)=x2−10x+mf(x)=x^2-10x+m, the graph has its vertex on the xx-axis. What is the value of mm?

  1. −25-25
  2. 00
  3. 2525 (correct answer)
  4. 5050
  5. 100100

Explanation: This question involves quadratic vertex properties. For f(x)=x² -10x + m, the vertex is at x= -(-10)/2=5, and f(5)=25 -50 + m = m-25. For vertex on x-axis, m-25=0, so m=25. Verifying, with m=25, f(x)=(x-5)², vertex at (5,0). This justifies m=25, choice C. A distractor like 0 might ignore the vertex calculation, setting m=0. Another, like 50, could double incorrectly.

Question 10

Which of the following statements is true about the quadratic y=ax2+bx+cy=ax^2+bx+c if a>0a>0 and c<0c<0?

  1. The graph opens downward.
  2. The graph has no real xx-intercepts.
  3. The graph must cross the xx-axis at exactly one point.
  4. The yy-intercept is below the xx-axis. (correct answer)
  5. The vertex must lie on the xx-axis.

Explanation: This question explores properties of quadratic graphs based on coefficients. For y = ax² + bx + c with a>0 and c<0, the parabola opens upward, and the y-intercept is c, which is negative. This means the graph crosses the y-axis below the x-axis. The sign of a determines the direction, and c directly gives the y-intercept. This justifies choice D as true. A distractor like choice A fails because a>0 means it opens upward, not downward, a sign misinterpretation. Another, like choice B, is not necessarily true since the discriminant could allow real intercepts.

Question 11

Which of the following statements is true about the polynomial q(x)=−2x2+7x−3q(x)=-2x^2+7x-3?

  1. The graph opens upward because the constant term is negative.
  2. The graph opens downward because the leading coefficient is negative. (correct answer)
  3. The graph must have no real zeros because the leading coefficient is negative.
  4. The yy-intercept is 77.
  5. The axis of symmetry is x=7x=7.

Explanation: This question examines properties of quadratic polynomials from coefficients. For q(x) = -2x² + 7x - 3, the leading coefficient -2 is negative, so the parabola opens downward. This determines the end behavior for large |x|. The y-intercept is -3, not 7, and axis is x = -7/(2*(-2)) = 7/4. This justifies choice B as true. A distractor like choice D fails because the y-intercept is the constant term -3, possibly confusing with the linear coefficient. Another, like choice A, errs on the sign of the leading coefficient.

Question 12

Which of the following represents the complete factorization of x2−9x+14x^2-9x+14 over the integers?

  1. (x−7)(x−2)(x-7)(x-2) (correct answer)
  2. (x+7)(x+2)(x+7)(x+2)
  3. (x−7)(x+2)(x-7)(x+2)
  4. (x−2)2(x-2)^2
  5. x(x−9)+14x(x-9)+14

Explanation: This question assesses factoring quadratic polynomials over the integers. To factor x² - 9x + 14, find two integers that multiply to 14 and add to -9, which are -7 and -2. Thus, it factors as (x-7)(x-2). Expanding (x-7)(x-2) gives x² - 2x - 7x + 14 = x² - 9x + 14, confirming the factorization. This complete factorization justifies choice A as correct. A representative distractor like (x+7)(x+2) fails because it expands to x² + 9x + 14, resulting from a sign error in the coefficients. Another distractor, (x-2)², expands to x² - 4x + 4, which does not match the linear term.

Question 13

Which of the following represents the complete factorization of x3−8x^3-8 over the integers?

  1. (x−2)(x2+2x+4)(x-2)(x^2+2x+4) (correct answer)
  2. (x+2)(x2−2x+4)(x+2)(x^2-2x+4)
  3. (x−2)3(x-2)^3
  4. (x−8)(x2+8x+64)(x-8)(x^2+8x+64)
  5. x(x2−8)x(x^2-8)

Explanation: This question tests factoring cubic polynomials over the integers using difference of cubes. The expression x³ - 8 = x³ - 2³ factors as (x-2)(x² + 2x + 4), per the formula a³ - b³ = (a-b)(a² + ab + b²). Expanding (x-2)(x² + 2x + 4) = x³ + 2x² + 4x - 2x² - 4x - 8 = x³ - 8 confirms. The quadratic x² + 2x + 4 has discriminant 4 - 16 = -12 <0, so no further real factors. This justifies choice A as complete. A distractor like (x+2)(x² - 2x + 4) expands to x³ + 8, a sign error in the formula. Another, like (x-2)³, does not match the constant term.

Question 14

If r(x)=(x−2)(x−5)r(x)=(x-2)(x-5), which of the following statements is true?

  1. r(x)r(x) has a zero at x=−2x=-2.
  2. r(x)r(x) has a zero at x=7x=7.
  3. r(x)r(x) is negative for all real xx.
  4. r(x)r(x) equals 00 when x=2x=2. (correct answer)
  5. The leading coefficient of r(x)r(x) is −1-1.

Explanation: This question assesses properties of quadratic polynomials in factored form. For r(x) = (x-2)(x-5), it equals zero at x=2 and x=5. Specifically, at x=2, (2-2)(2-5)=0*(-3)=0. The leading coefficient is positive 1, and it's positive outside [2,5], negative inside. This justifies choice D as true. A distractor like choice A claims zero at x=-2, but r(-2)=(-4)(-7)=28≠0, possibly from sign error. Another, like choice C, fails as it's not always negative.

Question 15

Which of the following represents the complete factorization of x2+2x−24x^2+2x-24 over the integers?

  1. (x+6)(x−4)(x+6)(x-4) (correct answer)
  2. (x−6)(x+4)(x-6)(x+4)
  3. (x+8)(x−3)(x+8)(x-3)
  4. (x+2)(x−12)(x+2)(x-12)
  5. (x+6)2−60(x+6)^2-60

Explanation: This question tests factoring quadratics over integers. For x² + 2x - 24, find integers multiplying to -24 and adding to 2: 6 and -4. Thus, (x+6)(x-4). Expanding: x² -4x +6x -24 = x² +2x -24 confirms. This is complete as factors are linear. This justifies choice A. A distractor like (x-6)(x+4)=x² -2x -24 fails due to sign error in the linear term. Another, like (x+2)(x-12)=x² -10x -24, mismatches the coefficient.

Question 16

Which of the following represents the complete factorization of x2−16x^2-16 over the integers?

  1. (x−8)(x+2)(x-8)(x+2)
  2. (x−4)(x+4)(x-4)(x+4) (correct answer)
  3. (x−4)2(x-4)^2
  4. x(x−16)x(x-16)
  5. (x2−4)(x2+4)(x^2-4)(x^2+4)

Explanation: This question tests factoring difference of squares in polynomials. x² - 16 = x² - 4² = (x-4)(x+4). Expanding (x-4)(x+4) = x² +4x -4x -16 = x² -16 confirms. This is complete over integers as both factors are linear. This justifies choice B. A distractor like (x-8)(x+2) = x² -6x -16 fails due to incorrect numbers that don't multiply to -16 or add properly, a factoring error. Another, like (x-4)², introduces a middle term.

Question 17

Let g(x)=x3−4x2−x+4g(x)=x^3-4x^2-x+4. Which of the following represents g(x)g(x) factored completely over the integers?

  1. (x−4)(x2−1)(x-4)(x^2-1)
  2. (x−1)(x+1)(x−4)(x-1)(x+1)(x-4) (correct answer)
  3. (x−1)(x2−4)(x-1)(x^2-4)
  4. (x+1)(x2−4x+4)(x+1)(x^2-4x+4)
  5. (x−1)(x+1)(x+4)(x-1)(x+1)(x+4)

Explanation: This question tests factoring a cubic polynomial completely. For g(x)=x3−4x2−x+4g(x)=x^3-4x^2-x+4, we can use factoring by grouping. Group the terms: (x3−4x2)+(−x+4)(x^3-4x^2)+(-x+4). Factor out common factors: x2(x−4)−1(x−4)=(x2−1)(x−4)x^2(x-4)-1(x-4)=(x^2-1)(x-4). Since x2−1x^2-1 is a difference of squares, it factors further as (x−1)(x+1)(x-1)(x+1). Therefore, g(x)=(x−1)(x+1)(x−4)g(x)=(x-1)(x+1)(x-4), which matches choice B. A common error would be to stop at the partial factorization (x−4)(x2−1)(x-4)(x^2-1) as in choice A, forgetting that x2−1x^2-1 can be factored further.