Rucete ✏ AP Precalculus In a Nutshell
2. Polynomial Functions — Practice Questions
This chapter drills polynomial degree/leading term, local/absolute extrema, points of inflection, zeros (real/complex, multiplicity), end behavior, equivalent forms (factoring/binomial theorem), and polynomial inequalities.
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(Multiple Choice — Click to Reveal Answer)
1. Which is a polynomial function?
(A) f(x)=x^(-1)+3x (B) g(x)=√x+1 (C) h(x)=x^3−4x+7 (D) p(x)=3/(x−2)
Answer
(C) — Only integer, nonnegative exponents are allowed for polynomials.
2. The degree and leading coefficient of p(x)=−7x^5+2x^3−4 are
(A) degree 3, leading 2 (B) degree 5, leading −7 (C) degree 5, leading 2 (D) degree 4, leading −7
Answer
(B) — Highest power is 5 with coefficient −7.
3. Which is written in standard (descending) order?
(A) x−2x^3+5 (B) 5−2x^3+x (C) −2x^3+x+5 (D) x+5−2x^3
Answer
(C) — Powers from high to low: −2x^3 + x + 5.
4. A constant function is a polynomial of degree
(A) 0 (B) 1 (C) undefined (D) depends on the constant
Answer
(A) — Nonzero constant has degree 0.
5. The graph of y=x^4 has a local extremum at
(A) x=−1 only (B) x=0 only (C) x=1 only (D) x=−1 and x=1
Answer
(B) — Minimum at x=0; elsewhere increasing.
6. A 5th-degree polynomial can have at most how many relative maxima/minima in total?
(A) 3 (B) 4 (C) 5 (D) 6
Answer
(B) — At most n−1 = 4 local extrema for degree n=5.
7. A 6th-degree polynomial can have at most how many points of inflection?
(A) 3 (B) 4 (C) 5 (D) 6
Answer
(B) — At most n−2 = 4 points of inflection for degree 6.
8. Even-degree polynomials (with nonzero leading term) have end behavior that
(A) rises left & falls right (B) falls left & rises right (C) both ends same direction (D) oscillates
Answer
(C) — Even degree: both ends up (positive lead) or both down (negative lead).
9. Odd-degree polynomials (with nonzero leading term) have end behavior that
(A) same direction both ends (B) opposite directions (C) always rises both ends (D) always falls both ends
Answer
(B) — Odd degree: one end up, the other down.
10. If a polynomial with real coefficients has a nonreal zero 2−5i, then it also has
(A) −2+5i (B) 2+5i (C) −2−5i (D) 5−2i
Answer
(B) — Complex zeros occur in conjugate pairs.
11. The graph of y=(x−3)^2 touches the x-axis at x=3 and is tangent there because the zero has
(A) odd multiplicity (B) even multiplicity (C) multiplicity 1 (D) multiplicity 0
Answer
(B) — Even multiplicity ⇒ tangent (“bounces”).
12. The zeros of p(x)=x(x+1)^2(x−4) are
(A) 0, −1, −1, 4 (B) 0, 1, 1, 4 (C) 0, −1, 4 (D) 0, 1, 4
Answer
(A) — −1 has multiplicity 2.
13. If p(x) has degree 4, then p(x) can have at most how many real x-intercepts?
(A) 1 (B) 2 (C) 3 (D) 4
Answer
(D) — At most degree many real zeros (counting multiplicities).
14. Which statement about absolute extrema on a closed interval [a,b] is true for polynomials?
(A) They may not exist (B) Must exist (Extreme Value Theorem) (C) Only maxima exist (D) Only minima exist
Answer
(B) — Continuous on [a,b] ⇒ max and min exist.
15. The y-intercept of p(x)=2x^3−3x^2+5 is
(A) (0,0) (B) (0,2) (C) (0,−3) (D) (0,5)
Answer
(D) — Evaluate at x=0.
16. If p(x) is even (p(−x)=p(x)), then its graph has symmetry about
(A) x-axis (B) y-axis (C) origin (D) line y=x
Answer
(B) — Even ⇒ y-axis symmetry.
17. If p(x) is odd (p(−x)=−p(x)), then its graph has symmetry about
(A) x-axis (B) y-axis (C) origin (D) line y=x
Answer
(C) — Odd ⇒ origin symmetry.
18. The leading term controls end behavior because
(A) lower-degree terms dominate as |x|→∞ (B) all terms cancel (C) highest power dominates growth (D) constant term dominates
Answer
(C) — Largest power grows fastest.
19. The degree of (x−1)^3(x+2)^2 is
(A) 3 (B) 4 (C) 5 (D) 6
Answer
(C) — Sum of multiplicities: 3+2=5.
20. The total number of complex zeros of a polynomial (counting multiplicity) equals
(A) its number of real zeros (B) its degree (C) its number of distinct factors (D) its number of terms
Answer
(B) — Fundamental Theorem of Algebra.
21. If p(x)=x^4−25, then its real zeros are
(A) ±5 (B) ±25 (C) ±5i (D) none
Answer
(A) — x^4−25=(x^2−5)(x^2+5) ⇒ real zeros ±√5? Wait: x^2−5=0 ⇒ x=±√5. 정정: 실제 실근은 ±√5. (선지 오기 가능 문제로 해설로 정정 표시)
22. If p(x)=x(x−4)^2(x+3), then at x=−3 the graph
(A) crosses (B) touches (C) has a hole (D) asymptote
Answer
(A) — Multiplicity 1 ⇒ crosses.
23. For g(x)=−2x^6+… the end behavior is
(A) rises both ends (B) falls both ends (C) falls left, rises right (D) rises left, falls right
Answer
(B) — Even degree, negative leading coefficient ⇒ both ends down.
24. For f(x)=3x^5+… the end behavior is
(A) rises left, falls right (B) falls left, rises right (C) rises both (D) falls both
Answer
(B) — Odd degree, positive lead: left down, right up.
25. The binomial coefficient “5 choose 2” equals
(A) 5 (B) 10 (C) 20 (D) 25
Answer
(B) — 5C2=10.
26. A 7th-degree polynomial can have at most how many turning points?
(A) 5 (B) 6 (C) 7 (D) 8
Answer
(B) — At most n−1 = 6 local extrema.
27. A 7th-degree polynomial can have at most how many points of inflection?
(A) 4 (B) 5 (C) 6 (D) 7
Answer
(B) — At most n−2 = 5.
28. Let p be a real-coefficient polynomial with zeros 2, −3, and 1−4i. The minimal possible degree is
(A) 3 (B) 4 (C) 5 (D) 6
Answer
(C) — Conjugate 1+4i must also be a zero ⇒ 4 zeros; but 2,−3 distinct ⇒ total 5? Wait: listed zeros are three (two real, one complex). With its conjugate, total distinct zeros = 4, so minimal degree 4. 정답: (B).
29. If p(x)=(x−1)^2(x+2)^3, then the graph at x=−2
(A) is tangent (B) crosses (C) has asymptote (D) has hole
Answer
(B) — Odd multiplicity (3) ⇒ crosses.
30. For f(x)=−x^4+5x^2−1, which statement is true?
(A) Even function, y-axis symmetry (B) Odd function, origin symmetry (C) Neither even nor odd (D) Linear
Answer
(A) — f(−x)=f(x).
31. The end behavior of h(x)=4(x−1)^8−7x^7+… is determined by
(A) −7x^7 (B) 4(x−1)^8 (C) constant term (D) middle terms
Answer
(A) — Highest power term −7x^7 dominates as |x|→∞.
32. Suppose a polynomial p has real coefficients and exactly two nonreal zeros. Which could be true?
(A) degree 3 with one real zero (B) degree 4 with one real zero (C) degree 5 with four nonreal zeros (D) degree 2 with two real zeros
Answer
(A) — Nonreal occur in conjugate pair; degree 3 ⇒ 1 real + 2 nonreal.
33. If x=1 is a zero of multiplicity 4 for p(x), then near x=1 the graph
(A) crosses sharply (B) crosses with flattening (C) touches and turns (very flat) (D) has vertical asymptote
Answer
(C) — Even multiplicity ⇒ touches; higher multiplicity ⇒ flatter.
34. The constant term of p(x)= (x^2−3x+2)(2x−5) equals
(A) −10 (B) 4 (C) −5 (D) 20
Answer
(A) — Constant term = (constant of first)*(constant of second) = 2*(−5)=−10.
35. Which inequality solution set matches f(x)=(x−4)^2(x+1)≤0 ?
(A) (−∞,−1]∪{4} (B) [−1,4] (C) [−1,∞) (D) (−∞,−1]∪[4,∞)
Answer
(B) — Sign chart: even at 4 (touch), odd at −1 (cross). Product ≤0 between −1 and 4 inclusive.
36. Find the y-intercept of p(x)=−3x^4+2x−7.
Answer
(0,−7) — Substitute x=0.
37. A degree-6 polynomial can have at most how many distinct real zeros?
Answer
6 — Cannot exceed the degree (counting multiplicity).
38. Give one polynomial of least degree with real coefficients having zeros 2, −1, and 3+i.
Answer
(x−2)(x+1)[(x−3)^2+1] — Include conjugate 3−i.
39. Factor completely over the reals: x^4−16.
Answer
(x^2−4)(x^2+4)=(x−2)(x+2)(x^2+4).
40. State the end behavior of p(x)=−5x^8+… using limits.
Answer
lim_{x→±∞} p(x)=−∞ — Even degree, negative lead.
41. Determine whether f(x)=x^3−x is even, odd, or neither.
Answer
Odd — f(−x)=−f(x).
42. Solve the inequality (x−2)(x+5)≥0 and express the solution set.
Answer
(−∞,−5]∪[2,∞) — Standard sign analysis.
43. A polynomial has zeros at x=−2 (mult 2) and x=3 (mult 1). Write one such polynomial with leading coefficient 1.
Answer
(x+2)^2(x−3).
44. Use the Binomial Theorem to write the x^3-term coefficient in (x+2)^5.
Answer
80 — 5C3 x^3 (2^2) = 10*4=40? Careful: term is 5C3 x^3 (2)^{2} ⇒ 10*4 = 40. 정정: 40.
45. Expand the first three terms (highest powers) of (2x−1)^4.
Answer
16x^4 − 32x^3 + 24x^2 + … — From binomial coefficients 1,4,6 with a=2x, b=−1.
46. For g(x)=x(x−1)(x+2), list the x-intercepts and state whether the graph crosses or touches at each.
Answer
(−2,0), (0,0), (1,0) — all multiplicity 1 ⇒ crosses at each.
47. Sketch logic: Give the possible degree and leading sign if a graph rises left and falls right and has 3 turning points.
Answer
Odd degree with negative leading coefficient; degree ≥4 (turning points ≤ n−1). Minimum degree 4 fits 3 turning points; odd-degree condition indicates actual degree could be 5 with negative lead (rises left, falls right).
48. Solve the inequality (x+1)^2(x−4) < 0.
Answer
(−1,4) — Even multiplicity at −1 (no sign change), cross at 4. Negative between −1 and 4.
49. Write a degree-5 polynomial (leading coeff 1) whose real zeros are −3 (mult 2), 0 (mult 1), and 4 (mult 2).
Answer
x(x+3)^2(x−4)^2.
50. If a polynomial p has zeros −2, 0, and 5 with p(1)=−12, find p(x).
Answer
Base form: k·x(x+2)(x−5). Evaluate at x=1: k·(1)(3)(−4)=−12 ⇒ k=1. p(x)=x(x+2)(x−5).