AIME Polynomials and Functions

Question 1

Factor completely: 2x^3 - 8x.

Accepted Answer

2x(x-2)(x+2)

Answer

Factor out 2x: 2x(x^2-4)=2x(x-2)(x+2).

Question 2

If P(x) = x^3 + 2x^2 - 5x - 6 and P(-1) = 0, fully factor P(x).

Accepted Answer

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

Answer

Since x=-1 is a root, (x+1) is a factor; dividing gives x^2+x-6=(x-2)(x+3).

Question 3

What is the degree of the product of x^3 + 1 and x^2 - x + 1?

Accepted Answer

5

Answer

The degree of a product equals the sum of degrees: 3+2=5.

Question 4

Find all rational roots of f(x) = x^3 - 6x^2 + 11x - 6.

Accepted Answer

{1, 2, 3}

Answer

By the Rational Root Theorem, test divisors of 6; x=1,2,3 all satisfy f(x)=0.

Question 5

If f(x) = 2x + 3, find f^(-1)(11).

Accepted Answer

4

Answer

Solve 2x+3=11: x=4, so f^(-1)(11)=4.

AIME - American Invitational Mathematics Examination Practice Test