AP PreCalculus Practice Quiz: Rational Functions and Zeros

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

What is the real zero of the rational function f(x) = (x - 8) / (x + 1)?

A)x = -1B)x = 1C)x = 8D)x = -8

All Questions (9)

What is the real zero of the rational function f(x) = (x - 8) / (x + 1)?

A) x = -1

B) x = 1

C) x = 8

D) x = -8

Correct Answer: C

The real zeros of a rational function correspond to the real zeros of the numerator, as long as those values are in the function's domain. Setting the numerator (x - 8) equal to zero gives x = 8. This value does not make the denominator zero, so it is the zero of the function.

For a rational function r(x) = p(x) / q(x), the real zeros are found by identifying the values of x for which...

A) q(x) = 0

B) p(x) = 0 and q(x) ≠ 0

C) p(x) = 0, regardless of the value of q(x)

D) p(x) and q(x) are both equal to zero

Correct Answer: B

The real zeros of a rational function are the real zeros of the numerator (p(x) = 0) for values that are in the function's domain. The domain excludes any values that make the denominator zero (q(x) ≠ 0).

What are the real zeros of the rational function g(x) = (x^2 - 9) / (x + 3)?

A) x = 3 and x = -3

B) x = 3 only

C) x = -3 only

D) The function has no real zeros.

Correct Answer: B

The zeros of the numerator, x^2 - 9, are x = 3 and x = -3. However, a zero of a rational function must be in its domain. The value x = -3 makes the denominator zero, so it is excluded from the domain. Therefore, the only real zero is x = 3.

When solving the inequality (x + 4) / (x - 2) ≥ 0, which values are used as endpoints to define the intervals for testing?

A) x = -4 only

B) x = 2 only

C) x = -4 and x = 2

D) x = 4 and x = -2

Correct Answer: C

To solve a rational inequality, the intervals to be tested are determined by the real zeros of both the numerator and the denominator. The zero of the numerator (x + 4) is x = -4, and the zero of the denominator (x - 2) is x = 2. These are the endpoints for the intervals.

Determine all the real zeros of the function h(x) = (x(x - 5)(x + 2)) / (x^2 + 4).

A) x = 0, x = 5, x = -2

B) x = 5, x = -2

C) x = 0 only

D) The function has no real zeros because the denominator is never zero.

Correct Answer: A

The real zeros of a rational function are the zeros of the numerator, provided they are in the domain. The numerator is zero at x = 0, x = 5, and x = -2. The denominator, x^2 + 4, is never zero for any real x, so the domain is all real numbers. Thus, all three values are zeros of the function.

For a rational function r(x), the real zeros of its numerator and denominator polynomials are significant when solving an inequality like r(x) ≤ 0 because these values are...

A) always the final solutions to the inequality.

B) the only points where the function's graph can cross the x-axis or have a vertical asymptote.

C) guaranteed to be excluded from the final solution set.

D) the locations of the function's maximum and minimum values.

Correct Answer: B

The real zeros of the numerator (the function's zeros) and the denominator (vertical asymptotes) are the critical points. They serve as endpoints for intervals because they are the only values where the function can change its sign from positive to negative or vice-versa.

Which of the following is a real zero of the function f(x) = (3x^2 + 5x - 2) / (x - 4)?

A) x = 2

B) x = -2

C) x = 4

D) x = -4

Correct Answer: B

To find the zeros, we set the numerator to zero: 3x^2 + 5x - 2 = 0. Factoring gives (3x - 1)(x + 2) = 0. The zeros of the numerator are x = 1/3 and x = -2. Since neither of these values makes the denominator zero, they are both real zeros of the function. Of the options provided, x = -2 is a correct answer.

Consider the rational function r(x) = (x^2 - 7x + 10) / (x^2 - 4). What are the real zeros of r(x)?

A) x = 2 and x = 5

B) x = 5 only

C) x = -2, x = 2, and x = 5

D) x = 2 only

Correct Answer: B

First, find the zeros of the numerator: x^2 - 7x + 10 = (x - 2)(x - 5) = 0, which gives x = 2 and x = 5. Next, check if these values are in the domain by finding the zeros of the denominator: x^2 - 4 = (x - 2)(x + 2) = 0, which gives x = 2 and x = -2. The value x = 2 is a zero of the numerator but is not in the domain of r(x). Therefore, the only real zero of the function is x = 5.

The set of numbers used to establish test intervals for solving the inequality r(x) ≤ 0 is composed of the real zeros of the numerator and the real zeros of the denominator. These numbers are collectively known as...

A) the solution set.

B) the domain of the function.

C) the endpoints or asymptotes that define the intervals.

D) the range of the function.

Correct Answer: C

This question is a direct application of the concept that the real zeros of both polynomial functions (numerator and denominator) of a rational function serve as the endpoints or asymptotes for the intervals that must be analyzed to solve a rational inequality.