AMC 12 Polynomials & Rational Functions 3

Question 1

If α and β are the roots of x² + px + q = 0, then α² + β² equals:

Accepted Answer

p² - 2q

Answer

α + β = -p and αβ = q, so α² + β² = (α + β)² - 2αβ = (-p)² - 2q = p² - 2q.

Question 2

The rational function f(x) = (x³ - x)/(x² - 1) simplifies to (for x ≠ ±1):

Accepted Answer

x

Answer

x³ - x = x(x² - 1), so (x³ - x)/(x² - 1) = x(x² - 1)/(x² - 1) = x.

Question 3

By the Rational Root Theorem, which of the following is a root of p(x) = 2x³ - 3x² - 11x + 6?

Accepted Answer

3

Answer

Possible rational roots are ±{1, 2, 3, 6}/±{1, 2}; testing x = 3: 2(27) - 3(9) - 11(3) + 6 = 54 - 27 - 33 + 6 = 0.

Question 4

Given that x = 3 is a root of p(x) = 2x³ - 3x² - 11x + 6, the complete factorization is:

Accepted Answer

(x - 3)(2x - 1)(x + 2)

Answer

Dividing by (x - 3) via synthetic division gives 2x² + 3x - 2, which factors as (2x - 1)(x + 2).

Question 5

The polynomial x⁴ + 4 factors over the integers as (Sophie Germain Identity):

Accepted Answer

(x² + 2x + 2)(x² - 2x + 2)

Answer

x⁴ + 4 = (x² + 2)² - (2x)² = (x² + 2x + 2)(x² - 2x + 2) using the difference of squares.

AMC 12 - - American Mathematics Contest 12 Practice Test