AMC 12 Polynomials & Rational Functions 1 — Questions and Answers
Question 1: By Vieta's formulas, the product of the roots of 3x⁴ - 2x³ + x² - 5x + 6 = 0 is:
- 2 (Correct answer)
- -2
- 3
- -3
Correct answer: 2
By Vieta's formulas, the product of all roots equals the constant term divided by the leading coefficient: 6/3 = 2.
Question 2: What is the remainder when p(x) = x¹⁰⁰ - 1 is divided by (x - 1)?
- 0 (Correct answer)
- 1
- -1
- 100
Correct answer: 0
By the Remainder Theorem, the remainder is p(1) = 1¹⁰⁰ - 1 = 0.
Question 3: The polynomial x⁴ + x² + 1 has how many real roots?
- 0 (Correct answer)
- 1
- 2
- 4
Correct answer: 0
Since x⁴ ≥ 0 and x² ≥ 0, we have x⁴ + x² + 1 ≥ 1 > 0 for all real x, so there are no real roots.
Question 4: If p(x) is a polynomial with p(2) = 5 and p(3) = 7, the remainder when p(x) is divided by (x-2)(x-3) is:
- 2x + 1 (Correct answer)
- 2x - 1
- x + 3
- 3x - 1
Correct answer: 2x + 1
The remainder R(x) = ax + b satisfies R(2) = 2a + b = 5 and R(3) = 3a + b = 7, giving a = 2, b = 1, so R(x) = 2x + 1.
Question 5: The polynomial x³ - 3x² + 3x - 1 = 0 has how many distinct positive real roots?
- 1 (Correct answer)
- 2
- 3
- 0
Correct answer: 1
x³ - 3x² + 3x - 1 = (x - 1)³, so x = 1 is the only root (with multiplicity 3).
Question 6: If f(x) = (x² - 4)/(x - 2) for x ≠ 2, then f(x) simplifies to:
- x + 2 (Correct answer)
- x - 2
- x² - 4
- x + 4
Correct answer: x + 2
x² - 4 = (x - 2)(x + 2), so (x² - 4)/(x - 2) = x + 2 for x ≠ 2.
Question 7: For the polynomial p(x) = xⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ with roots r₁, r₂, …, rₙ, by Vieta's formulas the sum r₁ + r₂ + … + rₙ equals:
- -aₙ₋₁ (Correct answer)
- aₙ₋₁
- a₀
- -a₀
Correct answer: -aₙ₋₁
For a monic polynomial, Vieta's formulas give the sum of roots as the negative of the coefficient of xⁿ⁻¹.
AMC 12 Polynomials & Rational Functions 1
By Vieta's formulas, the product of the roots of 3x⁴ - 2x³ + x² - 5x + 6 = 0 is: