Rucete ✏ AP Precalculus In a Nutshell
2. Polynomial Functions — Practice Questions 2
This chapter explores the properties, graphs, and applications of polynomial functions.
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(Multiple Choice — Click to Reveal Answer)
1. Which of the following is a polynomial function?
(A) f(x) = √x + 2
(B) f(x) = 3/x + 1
(C) f(x) = x³ - 2x² + 5
(D) f(x) = ln(x)
Answer
C — A polynomial must have whole number exponents and no variables in denominators or radicals.2. What is the degree of f(x) = 7x⁵ - 4x² + 6?
(A) 2
(B) 5
(C) 6
(D) 7
Answer
B — The highest power of x is 5, so the degree is 5.3. The leading coefficient of f(x) = -3x⁴ + 2x³ - x + 7 is:
(A) -3
(B) 4
(C) 2
(D) 7
Answer
A — The coefficient of the highest degree term (-3x⁴) is -3.4. Which function has end behavior rising left and rising right?
(A) f(x) = -2x³
(B) f(x) = x² + 3
(C) f(x) = -x⁴
(D) f(x) = 5x³ - 7
Answer
B — Even degree with positive leading coefficient → rises both ends.5. A 4th-degree polynomial has at most how many relative extrema?
(A) 1
(B) 2
(C) 3
(D) 4
Answer
C — A degree n polynomial has at most n-1 extrema, so 3 for degree 4.6. Which polynomial has odd degree?
(A) f(x) = x⁶ - 2x⁴
(B) g(x) = -5x⁷ + 3x²
(C) h(x) = x² + 1
(D) k(x) = 2x⁴ + 7
Answer
B — Degree is 7, which is odd.7. What is the constant term of f(x) = 3x⁵ - 7x² + 9?
(A) 3
(B) -7
(C) 5
(D) 9
Answer
D — The constant term is the standalone number, 9.8. Which of the following has exactly 3 real zeros?
(A) f(x) = (x - 1)(x² + 4)
(B) g(x) = (x² + 9)(x² + 1)
(C) h(x) = (x - 2)(x - 3)(x - 4)
(D) k(x) = (x + 1)²(x² + 1)
Answer
C — Three distinct linear factors → three real zeros.9. If a polynomial is tangent to the x-axis at x = -2, the root has:
(A) Odd multiplicity
(B) Even multiplicity
(C) Multiplicity of 1
(D) No multiplicity
Answer
B — Tangency at an x-intercept means even multiplicity.10. The polynomial f(x) = x³ - 4x² + 4x has a factor of:
(A) x + 4
(B) x - 2
(C) x² + 4
(D) x + 2
Answer
B — Factoring: f(x) = x(x² - 4x + 4) = x(x - 2)².11. Which describes the end behavior of f(x) = -2x⁶ + 3x²?
(A) Falls left, rises right
(B) Falls left, falls right
(C) Rises left, falls right
(D) Rises left, rises right
Answer
B — Even degree with negative leading coefficient → falls both ends.12. A 5th-degree polynomial can have at most how many inflection points?
(A) 2
(B) 3
(C) 4
(D) 5
Answer
B — Maximum inflection points = n - 2 = 3.13. Which polynomial has symmetry about the y-axis?
(A) f(x) = x³ - x
(B) g(x) = x² + 4
(C) h(x) = x⁵ - 2
(D) k(x) = x³ + 7x²
Answer
B — Even power terms only → even function (y-axis symmetry).14. The zeros of f(x) = (x - 1)(x + 1)(x - 3) are:
(A) 1, -1, -3
(B) 1, -1, 3
(C) 1, 0, -3
(D) -1, 0, 3
Answer
B — Set each factor equal to zero → 1, -1, 3.15. If f(x) = x⁴ - 16, which method factors it?
(A) Grouping
(B) Sum of cubes
(C) Difference of squares
(D) None
Answer
C — x⁴ - 16 = (x² - 4)(x² + 4) = (x - 2)(x + 2)(x² + 4).16. Which root must exist if 3 - 2i is a zero of a polynomial with real coefficients?
(A) 3 + 2i
(B) -3 - 2i
(C) 2 - 3i
(D) -2 + 3i
Answer
A — Complex roots appear in conjugate pairs.17. A polynomial of degree 7 must have exactly how many zeros (counting multiplicity)?
(A) At most 7
(B) At least 7
(C) Exactly 7
(D) Cannot be determined
Answer
C — Fundamental Theorem of Algebra: degree n polynomial has exactly n zeros (real + complex).18. Which graph shows end behavior: rises left, falls right?
(A) Positive odd-degree
(B) Negative odd-degree
(C) Positive even-degree
(D) Negative even-degree
Answer
B — Odd degree with negative leading coefficient.19. Which polynomial has degree 6?
(A) f(x) = (x² + 1)(x⁴ - x)
(B) g(x) = (x³ + 1)(x² + 1)
(C) h(x) = (x²)(x³)(x)
(D) k(x) = (x - 1)(x - 2)(x - 3)
Answer
A — Multiplying gives x⁶ + … so degree 6.20. The y-intercept of f(x) = 2x³ - 5x + 4 is:
(A) (0, -5)
(B) (0, 4)
(C) (0, 2)
(D) (0, -4)
Answer
B — Substitute x = 0 → f(0) = 4.21. Which factorization is correct for f(x) = x³ + 8?
(A) (x + 2)(x² - 2x + 4)
(B) (x - 2)(x² + 2x + 4)
(C) (x + 2)(x² + 2x + 4)
(D) (x - 2)(x² - 2x + 4)
Answer
C — Sum of cubes: x³ + 2³ = (x + 2)(x² - 2x + 4).22. Which root has multiplicity 2 in f(x) = (x - 1)²(x + 3)?
(A) -3
(B) 1
(C) 0
(D) 2
Answer
B — The squared factor indicates multiplicity 2 at x = 1.23. If a polynomial has 4 real zeros, its degree is at least:
(A) 2
(B) 3
(C) 4
(D) 5
Answer
C — To have 4 distinct real zeros, degree ≥ 4.24. Which function is odd?
(A) f(x) = x² + 2
(B) g(x) = -x³
(C) h(x) = x⁴ - 5
(D) k(x) = x² - x⁴
Answer
B — f(-x) = -f(x) → odd function.25. The binomial expansion of (x + 1)² is:
(A) x² + 2x + 1
(B) x² - 2x + 1
(C) x² + 1
(D) x² - 1
Answer
A — (x + 1)(x + 1) = x² + 2x + 1.26. Which of the following could be the graph of f(x) = (x - 2)²(x + 1)?
(A) Crosses x-axis at both -1 and 2
(B) Crosses at -1 and bounces at 2
(C) Bounces at -1 and crosses at 2
(D) Tangent at both -1 and 2
Answer
B — Multiplicity 1 at x = -1 (crosses), multiplicity 2 at x = 2 (bounces).27. If f(x) = x⁴ - 5x² + 4, which is a factor?
(A) (x - 1)
(B) (x - 2)
(C) (x² - 4)
(D) (x² + 5)
Answer
C — Factorization: (x² - 1)(x² - 4).28. Which describes the end behavior of f(x) = -3x⁵ + x³?
(A) Falls left, rises right
(B) Rises left, falls right
(C) Falls both ends
(D) Rises both ends
Answer
B — Odd degree, negative leading coefficient → rises left, falls right.29. If 2 + i is a root of a polynomial with real coefficients, which must also be a root?
(A) 2 - i
(B) -2 + i
(C) -2 - i
(D) 2 + i
Answer
A — Complex roots occur in conjugate pairs.30. A polynomial of degree 6 can have at most how many turning points?
(A) 4
(B) 5
(C) 6
(D) 7
Answer
B — Maximum turning points = n - 1 = 5.31. Which polynomial has a root of multiplicity 3?
(A) f(x) = (x - 1)(x + 2)
(B) g(x) = (x - 4)³(x² + 1)
(C) h(x) = (x² - 9)(x - 5)
(D) k(x) = (x + 3)(x - 3)²
Answer
B — (x - 4)³ indicates multiplicity 3.32. The binomial expansion of (x - 2)³ includes which middle term?
(A) -8x²
(B) 12x²
(C) -12x²
(D) 6x²
Answer
C — Expansion: x³ - 6x² + 12x - 8 → middle term is -6x².33. Which polynomial has degree 5 and positive leading coefficient?
(A) f(x) = -2x⁵ + 3x³
(B) g(x) = x(x - 1)²(x + 2)²
(C) h(x) = (x² + 1)(x³ - 4)
(D) Both B and C
Answer
D — Both expand to degree 5, leading coefficient positive.34. If f(x) = (x - 1)(x + 2)(x - 3), the sum of zeros is:
(A) -2
(B) 0
(C) 2
(D) -4
Answer
C — Sum of zeros = 1 - 2 + 3 = 2.35. Which inequality is satisfied by f(x) = (x - 1)(x + 2)² ≥ 0?
(A) x ≤ -2 or x ≥ 1
(B) x ≥ -2 and x ≤ 1
(C) x ≥ -2 only
(D) All real x
Answer
A — Sign chart: positive left of -2, negative between -2 and 1, positive after 1. At -2 root multiplicity even → stays ≥ 0.36. Find the degree of f(x) = (x² - 1)(x - 3)(x + 2).
Answer
Degree = 4 (2 + 1 + 1).37. Write f(x) = x³ - 9x factored form.
Answer
x(x - 3)(x + 3).38. Determine the leading coefficient of f(x) = (2x - 1)(x² + 3).
Answer
Leading term = 2x³, so coefficient = 2.39. If a polynomial is even, what symmetry does its graph have?
Answer
Y-axis symmetry.40. If a polynomial is odd, what symmetry does its graph have?
Answer
Origin symmetry.41. Solve f(x) = x³ - 4x = 0.
Answer
x(x - 2)(x + 2) = 0 → zeros: -2, 0, 2.42. Find the constant term in (x - 2)⁴.
Answer
(-2)⁴ = 16.43. What is the y-intercept of f(x) = x⁴ - 3x² + 2?
Answer
(0, 2).44. Determine the multiplicity of zero at x = -3 for f(x) = (x + 3)⁴(x - 1).
Answer
Multiplicity = 4.45. Factor completely: x⁴ - 81.
Answer
(x² - 9)(x² + 9) = (x - 3)(x + 3)(x² + 9).46. If f(x) = 2x³ - 5x² + x - 4, what is the degree?
Answer
Degree = 3.47. Give an example of a polynomial with degree 2 and leading coefficient -1.
Answer
f(x) = -x² + 3x + 2.48. Find the end behavior of f(x) = -x⁶ + 2x³.
Answer
Falls left, falls right (even degree, negative leading coefficient).49. If f(x) = (x - 1)(x + 1)(x² + 1), how many real zeros?
Answer
2 real zeros (x = ±1).50. Expand (x + y)³.