AP PreCalculus Practice Quiz: Polynomial Functions and Complex Zeros

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

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

A polynomial function p(x) has a degree of 7. According to the fundamental theorem of algebra, how many complex zeros does p(x) have, counting multiplicities?

A)At most 7B)Exactly 7C)At least 7D)It cannot be determined from the degree

All Questions (16)

A polynomial function p(x) has a degree of 7. According to the fundamental theorem of algebra, how many complex zeros does p(x) have, counting multiplicities?

A) At most 7

B) Exactly 7

C) At least 7

D) It cannot be determined from the degree

Correct Answer: B

Based on the provided content, a polynomial function of degree n has exactly n complex zeros when counting multiplicities. Since the degree is 7, there must be exactly 7 complex zeros.

If a polynomial function p(x) with real coefficients has a non-real zero of 3 + 2i, which of the following must also be a zero of p(x)?

A) -3 + 2i

B) -3 - 2i

C) 3 - 2i

D) 2 + 3i

Correct Answer: C

The content states that if a+bi is a non-real zero of a polynomial function, then its conjugate, a-bi, is also a zero. The conjugate of 3 + 2i is 3 - 2i.

A function f(x) is an odd function. Which of the following analytical properties must be true for all x in the domain of f?

A) f(-x) = f(x)

B) f(-x) = -f(x)

C) f(x) = |f(x)|

D) f(x) = f(x+1)

Correct Answer: B

The provided content explicitly states that an odd function analytically has the property f(-x) = -f(x).

The graph of a polynomial function p(x) is shown. At the real zero x = 2, the graph touches the x-axis but does not cross it. What can be concluded about the multiplicity of the zero at x = 2?

A) The multiplicity is odd.

B) The multiplicity is even.

C) The multiplicity is 1.

D) The multiplicity is non-real.

Correct Answer: B

The content states that if a real zero has even multiplicity, the signs of the output values are the same for input values near that zero. This corresponds to the graph touching the x-axis at the zero without crossing it. Therefore, the multiplicity must be even.

A polynomial function is defined by p(x) = (x - 4)^3(x^2 + 9). How many distinct complex zeros does p(x) have?

A) 2

B) 3

C) 4

D) 5

Correct Answer: B

The zeros can be found by setting each factor to zero. From (x-4)^3 = 0, we get the real zero x = 4. From x^2 + 9 = 0, we get x^2 = -9, which gives the non-real zeros x = 3i and x = -3i. Therefore, there are three distinct complex zeros: 4, 3i, and -3i.

A polynomial function with real coefficients is known to have zeros at x = 5, x = 2i, and x = 1-i. What is the minimum possible degree of this polynomial?

A) 3

B) 4

C) 5

D) 6

Correct Answer: C

Since the polynomial has real coefficients, non-real zeros must come in conjugate pairs. The zero 2i implies its conjugate, -2i, is also a zero. The zero 1-i implies its conjugate, 1+i, is also a zero. The real zero is x=5. This gives a minimum of 5 zeros: 5, 2i, -2i, 1-i, and 1+i. A polynomial with 5 zeros must have a degree of at least 5.

If the complex number c is a root of the polynomial equation p(x) = 0, which of the following statements is true?

A) p(c) = 0

B) p(0) = c

C) c must be a real number

D) The degree of p(x) is c

Correct Answer: A

The definition provided states that if a complex number 'a' is a zero of the polynomial function p, or a root of p(x) = 0, then p(a) = 0. Therefore, if c is a root, p(c) = 0.

Which of the following describes the graphical symmetry of an odd function?

A) Symmetry about the y-axis

B) Symmetry about the x-axis

C) Symmetry about the point (0,0)

D) Symmetry about the line y = x

Correct Answer: C

The content specifies that an odd function is graphically symmetric about the point (0,0), which is also known as symmetry with respect to the origin.

Consider a polynomial function p(x). For input values just less than 3, p(x) is positive. For input values just greater than 3, p(x) is also positive. If p(3) = 0, what can be inferred about the zero at x = 3?

A) The zero at x = 3 has odd multiplicity.

B) The zero at x = 3 has even multiplicity.

C) The zero at x = 3 is a non-real zero.

D) The polynomial is an even function.

Correct Answer: B

The content states that if a real zero 'a' has even multiplicity, then the signs of the output values are the same for input values near x = a. Since the output values are positive on both sides of x = 3, the multiplicity of this zero must be even.

A fifth-degree polynomial function, p(x), with real coefficients has known zeros at x = -2 (multiplicity 2) and x = 4i. What are the remaining zeros of p(x)?

A) -4i

B) -4i and another real zero

C) 4i (multiplicity 2) and -4i

D) 2 and -4i

Correct Answer: B

A fifth-degree polynomial must have 5 complex zeros. We are given x = -2 with multiplicity 2, which accounts for two zeros. We are given the non-real zero x = 4i. Because the coefficients are real, its conjugate, x = -4i, must also be a zero. This accounts for a total of 2 + 1 + 1 = 4 zeros. Since the degree is 5, there must be one more zero. This fifth zero must be real, because if it were non-real, its conjugate would also have to be a zero, making a total of 6 zeros, which is impossible for a degree 5 polynomial.

Which of the following polynomial functions could be an even function?

A) f(x) = x^3 + x

B) f(x) = x^4 + 2x^2 + 5

C) f(x) = x^5 - 3x^3 + x - 1

D) f(x) = x^2 + x

Correct Answer: B

An even function has the property f(-x) = f(x). A polynomial function is even if all of its terms have even-powered variables (including the constant term, which can be written as 5x^0). In option B, f(-x) = (-x)^4 + 2(-x)^2 + 5 = x^4 + 2x^2 + 5 = f(x). The other options contain odd-powered terms, which prevent them from being even functions.

The output values of a polynomial function p(x) for equal-interval input values are analyzed. The first differences are not constant. The second differences are not constant. The third differences are all equal to 12. What is the degree of the polynomial function p(x)?

A) 2

B) 3

C) 4

D) 12

Correct Answer: B

The content states that the degree of a polynomial can be found by examining successive differences. If the nth differences are the first set of constant differences, then the degree of the polynomial is n. Since the third differences are constant, the degree of the polynomial is 3.

The point (3, -5) is on the graph of an odd function, f(x). Which of the following points must also be on the graph of f(x)?

A) (-3, -5)

B) (3, 5)

C) (-3, 5)

D) (-5, 3)

Correct Answer: C

An odd function has the property f(-x) = -f(x). If the point (3, -5) is on the graph, it means f(3) = -5. Using the property, f(-3) = -f(3) = -(-5) = 5. Therefore, the point (-3, 5) must also be on the graph.

A polynomial p(x) has a degree of 4. Which of the following describes the number and nature of its zeros?

A) Exactly 4 real zeros.

B) Exactly 4 non-real zeros.

C) Exactly 4 complex zeros, which may be real or non-real.

D) At most 4 complex zeros.

Correct Answer: C

A polynomial function of degree n has exactly n complex zeros. Real numbers are a subset of complex numbers (e.g., 5 can be written as 5+0i). Therefore, a degree 4 polynomial has exactly 4 complex zeros, but these zeros can be a combination of real and non-real numbers.

A table of values for a polynomial function f(x) is given. What is the degree of f(x)? x: 0, 1, 2, 3, 4 f(x): 2, 3, 10, 29, 66

A) 2

B) 3

C) 4

D) 5

Correct Answer: B

We must find the successive differences. First differences: (3-2)=1, (10-3)=7, (29-10)=19, (66-29)=37. Second differences: (7-1)=6, (19-7)=12, (37-19)=18. Third differences: (12-6)=6, (18-12)=6. Since the third differences are constant, the degree of the polynomial is 3.

A polynomial function p(x) with real coefficients has a non-real zero a+bi. Which of the following statements is a direct consequence?

A) The polynomial must have an odd degree.

B) The value p(a-bi) must equal 0.

C) The value p(a) must be 0.

D) The polynomial cannot have any real zeros.

Correct Answer: B

The content provides two key rules. First, if a+bi is a zero, then p(a+bi) = 0. Second, if a+bi is a non-real zero of a polynomial with real coefficients, its conjugate a-bi is also a zero. If a-bi is a zero, then by definition, p(a-bi) must equal 0.