AMC 12 Polynomials & Rational Functions

Question 1

By Vieta's formulas, the product of the roots of 3x⁴ - 2x³ + x² - 5x + 6 = 0 is:

Accepted Answer

2

Answer

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)?

Accepted Answer

0

Answer

By the Remainder Theorem, the remainder is p(1) = 1¹⁰⁰ - 1 = 0.

Question 3

The polynomial x⁴ + x² + 1 has how many real roots?

Accepted Answer

0

Answer

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:

Accepted Answer

2x + 1

Answer

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?

Accepted Answer

1

Answer

x³ - 3x² + 3x - 1 = (x - 1)³, so x = 1 is the only root (with multiplicity 3).

AMC 12 - - American Mathematics Contest 12 Practice Test