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

p² + 2q

Answer

p² - q

Answer

-p² + 2q

Question 2

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

Accepted Answer

x

Answer

x - 1

Answer

x + 1

Answer

x²

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

-3

Answer

6

Answer

-6

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

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

Answer

(x - 3)(x - 1)(2x + 6)

Answer

(x + 3)(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

(x² + 2)²

Answer

(x² + 2x - 2)(x² - 2x - 2)

Answer

(x² + 4)(x² - 1)

Question 6

If a degree-4 polynomial with real coefficients has roots 2, -1, and 3 + i, what must the fourth root be?

Accepted Answer

3 - i

Answer

3 + i

Answer

-3 + i

Answer

-3 - i

Question 7

The number of sign changes in p(x) = x⁵ - 3x⁴ + x³ + 2x² - x + 1 indicates, by Descartes' Rule of Signs, at most how many positive real roots?

Accepted Answer

4

Answer

5

Answer

3

Answer

2