Rucete ✏ AP Precalculus In a Nutshell
2. Polynomial Functions — Practice Questions 3
This chapter explores the properties of polynomial functions, including zeros, multiplicities, extrema, end behavior, and inequalities.
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(Multiple Choice — Click to Reveal Answer)
1. Which of the following is NOT a polynomial function?
A. f(x) = 3x⁴ - 2x + 7
B. g(x) = x³ + √x
C. h(x) = 5
D. p(x) = -2x² + x - 1
Answer
B. g(x) = x³ + √x is not a polynomial because √x has a fractional exponent.2. The degree of the polynomial f(x) = (x - 2)(x + 3)(x - 1)(x + 4) is:
A. 2
B. 3
C. 4
D. 5
Answer
C. The polynomial has 4 linear factors, so its degree is 4.3. The leading coefficient of f(x) = -7x⁵ + 2x³ - x + 9 is:
A. -7
B. 5
C. 9
D. -1
Answer
A. The coefficient of the highest degree term (-7x⁵) is -7.4. How many complex zeros (real and non-real) does a polynomial of degree 6 have?
A. At most 6
B. Exactly 6
C. At least 6
D. Cannot be determined
Answer
B. By the Fundamental Theorem of Algebra, a degree 6 polynomial has exactly 6 complex zeros (counting multiplicity).5. Which statement is true about a polynomial of odd degree with a positive leading coefficient?
A. The graph rises left and rises right
B. The graph falls left and rises right
C. The graph rises left and falls right
D. The graph falls left and falls right
Answer
B. Odd degree with positive leading coefficient: left → down, right → up.6. If f(x) = (x - 3)²(x + 1), which best describes the behavior at x = 3?
A. Crosses the x-axis
B. Touches the x-axis and turns
C. Has a vertical asymptote
D. No intercept
Answer
B. A zero of even multiplicity (2) causes the graph to touch and turn.7. The maximum number of relative extrema of a degree 7 polynomial is:
A. 6
B. 7
C. 8
D. Cannot be determined
Answer
A. At most n - 1 = 6 relative extrema for degree 7.8. Which polynomial is even?
A. f(x) = x³ - 2x
B. g(x) = x⁴ - x² + 1
C. h(x) = x⁵ + 1
D. p(x) = x³ + x
Answer
B. Even powers only → symmetric about the y-axis.9. Which polynomial is odd?
A. f(x) = x² + 1
B. g(x) = x³ - x
C. h(x) = x⁴ - 2
D. p(x) = 2x² - x
Answer
B. Odd function satisfies f(-x) = -f(x).10. The polynomial f(x) = (x - 2)(x + 2)(x - 5) has how many real zeros?
A. 1
B. 2
C. 3
D. 4
Answer
C. The three linear factors give three real zeros: -2, 2, and 5.11. If p(x) = x³ - 4x² + 5x - 2, the constant term is:
A. -2
B. 5
C. -4
D. 3
Answer
A. The constant term is -2.12. A degree 5 polynomial must have at least how many turning points?
A. 0
B. 1
C. 4
D. 5
Answer
A. It may have as few as 0 turning points, maximum n-1 = 4.13. Which binomial expansion begins with x⁴ - 4x³y + 6x²y² ... ?
A. (x - y)³
B. (x - y)⁴
C. (x + y)⁴
D. (x - 2y)²
Answer
B. (x - y)⁴ expands to x⁴ - 4x³y + 6x²y² - 4xy³ + y⁴.14. The sum of the multiplicities of all zeros of a polynomial equals:
A. The constant term
B. The degree
C. The number of distinct real zeros
D. The number of factors
Answer
B. By definition, the sum of multiplicities equals the degree.15. If f(x) = (x - 1)³(x + 2), then f has degree:
A. 3
B. 4
C. 5
D. 6
Answer
B. Total degree = 3 + 1 = 4.16. The end behavior of f(x) = -2x⁶ + 5x⁴ - x is:
A. Rises left, rises right
B. Falls left, rises right
C. Falls left, falls right
D. Rises left, falls right
Answer
C. Even degree, negative leading coefficient → both ends down.17. Which of the following could be the graph of y = (x - 2)²(x + 1)?
A. Crosses at -1 and touches at 2
B. Crosses at 2 and -1
C. Tangent at -1
D. Vertical asymptotes at -1 and 2
Answer
A. Odd multiplicity crosses, even multiplicity touches.18. Which theorem guarantees a polynomial has both a max and min on a closed interval?
A. Fundamental Theorem of Algebra
B. Extreme Value Theorem
C. Intermediate Value Theorem
D. Local Extrema Theorem
Answer
B. Extreme Value Theorem applies to continuous functions on closed intervals.19. The graph of a polynomial of degree 4 with positive leading coefficient will:
A. Rise left, rise right
B. Fall left, rise right
C. Rise left, fall right
D. Fall left, fall right
Answer
A. Even degree, positive coefficient → both ends rise.20. The binomial coefficient C(6, 2) equals:
A. 12
B. 15
C. 20
D. 30
Answer
B. 6C2 = 6×5/2 = 15.21. If f(x) = (x - 3)(x + 3)(x - 1), what is the y-intercept?
A. -9
B. 3
C. -27
D. 9
Answer
C. f(0) = (-3)(3)(-1) = 9, correction: the answer is D (9).22. If f(x) = (x - 1)(x - 2)(x - 3)(x - 4), how many x-intercepts does it have?
A. 1
B. 2
C. 3
D. 4
Answer
D. Four linear factors → four intercepts.23. Which of the following statements about odd functions is true?
A. Graph is symmetric about the y-axis
B. f(-x) = f(x)
C. Graph is symmetric about the origin
D. Always increasing
Answer
C. Odd functions are symmetric about the origin.24. Which is the constant term in (x + 2)³?
A. 2
B. 6
C. 8
D. 12
Answer
C. (x + 2)³ = x³ + 6x² + 12x + 8, constant = 8.25. Which polynomial has exactly 2 distinct real zeros?
A. (x - 1)²(x - 2)²
B. (x - 1)(x² + 1)
C. (x - 2)(x - 3)(x - 4)
D. (x² + 1)(x² + 4)
Answer
A. Only x = 1 and x = 2 are real, each with multiplicity 2.26. A polynomial of degree 8 can have at most how many turning points?
A. 6
B. 7
C. 8
D. 9
Answer
B. At most n - 1 = 7 turning points.27. If f(x) = (x - 2)³(x + 1), the behavior at x = -1 is:
A. Crosses through
B. Touches and turns
C. No intercept
D. Vertical asymptote
Answer
A. Multiplicity 1 (odd) → crosses the axis.28. Which polynomial must have symmetry about the y-axis?
A. f(x) = x³ - 4x
B. g(x) = x⁴ + 2x² + 5
C. h(x) = x⁵ + 2x²
D. p(x) = 2x³ + 7
Answer
B. Even powers only → even function.29. The polynomial f(x) = x⁴ - 5x² + 6 can be factored as:
A. (x² - 2)(x² - 3)
B. (x - 2)(x + 2)(x - 3)(x + 3)
C. (x² - 6)(x² + 1)
D. Prime
Answer
A. Substitution y = x² → (y - 2)(y - 3).30. Which statement is true about zeros of real polynomials?
A. Complex roots appear singly
B. Complex roots appear in conjugate pairs
C. Complex roots never exist
D. Real roots must all be positive
Answer
B. If nonreal complex zeros exist, they occur as conjugate pairs.31. If a polynomial has degree 5, then the maximum number of distinct real zeros is:
A. 3
B. 4
C. 5
D. 6
Answer
C. A degree 5 polynomial can have at most 5 real zeros.32. The expansion of (x - 2y)³ begins with:
A. x³ - 6x²y + 12xy²
B. x³ + 2x²y - 4xy²
C. x³ - 2x²y + 4xy²
D. x³ + 6x²y + 12xy²
Answer
A. Using binomial theorem, (x - 2y)³ = x³ - 6x²y + 12xy² - 8y³.33. Which of the following could be the graph of an odd degree polynomial with negative leading coefficient?
A. Rises left, rises right
B. Falls left, rises right
C. Rises left, falls right
D. Falls left, falls right
Answer
C. Odd degree, negative coefficient: left up, right down.34. If p(x) = (x² + 1)(x - 3)(x + 2), then how many real zeros does it have?
A. 1
B. 2
C. 3
D. 4
Answer
C. Two from linear factors (x = 3, -2) and none from x² + 1.35. Which of the following polynomials has x = 0 as a root of multiplicity 4?
A. x⁴ - 16
B. x²(x² + 1)
C. x³(x - 1)
D. (x² - 4)(x² + 4)
Answer
B. x²(x² + 1) → x = 0 with multiplicity 2, correction: none here. Proper is x⁴.Short-Answer Questions
36. Find the degree and leading coefficient of f(x) = -2(x - 1)(x + 3)(x - 5).
Answer
Degree 3, leading coefficient -2.37. Determine the end behavior of f(x) = 5x⁷ - x³ + 2.
Answer
Odd degree, positive coefficient → falls left, rises right.38. State the number of possible real zeros of a degree 10 polynomial.
Answer
Maximum 10 real zeros, could be fewer.39. Factor completely: f(x) = x³ - 27.
Answer
(x - 3)(x² + 3x + 9), difference of cubes.40. If f(x) = (x - 2)²(x + 4), what is the multiplicity of the zero at x = 2?
Answer
Multiplicity 2 (even).41. What is the constant term of (x - 5)³?
Answer
-125.42. Find the y-intercept of f(x) = (x + 1)(x - 2)(x + 3).
Answer
f(0) = (1)(-2)(3) = -6.43. If a polynomial is even, describe its symmetry.
Answer
Symmetric about the y-axis.44. If a polynomial is odd, describe its symmetry.
Answer
Symmetric about the origin.45. Find all real zeros of f(x) = x² - 9.
Answer
x = -3, 3.46. Factor completely: f(x) = x⁴ - 16.
Answer
(x² - 4)(x² + 4) = (x - 2)(x + 2)(x² + 4).47. Determine the possible number of turning points of a degree 9 polynomial.
Answer
At most 8 turning points.48. Using the Intermediate Value Theorem, explain why f(x) = x³ - x - 2 has a real root between x = 1 and x = 2.
Answer
f(1) = -2 (negative), f(2) = 4 (positive), so a root exists between.49. Write the polynomial of least degree with real coefficients that has zeros at 2, -2, and i.
Answer
(x - 2)(x + 2)(x - i)(x + i) = (x² - 4)(x² + 1).50. Expand (x + y)⁴ up to the third term.