AIME Polynomials and Functions 1 — Questions and Answers
Question 1: Factor completely: 2x^3 - 8x.
- 2x(x-2)(x+2) (Correct answer)
- 2x(x^2-4)
- x(2x^2-8)
- 2(x^3-4x)
Correct answer: 2x(x-2)(x+2)
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).
- (x+1)(x-2)(x+3) (Correct answer)
- (x-1)(x+2)(x-3)
- (x+1)(x+2)(x-3)
- (x-1)(x-2)(x+3)
Correct answer: (x+1)(x-2)(x+3)
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?
- 5 (Correct answer)
- 6
- 4
- 3
Correct answer: 5
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.
- {1, 2, 3} (Correct answer)
- {1, 2, 6}
- {1, 3, 6}
- {2, 3, 6}
Correct answer: {1, 2, 3}
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).
- 4 (Correct answer)
- 25
- 8
- 7
Correct answer: 4
Solve 2x+3=11: x=4, so f^(-1)(11)=4.
Question 6: What is the sum of the roots of 4x^3 - 8x^2 + 5x - 1 = 0?
- 2 (Correct answer)
- 5/4
- 1/4
- 8
Correct answer: 2
By Vieta's formulas, sum of roots = -(-8)/4 = 2.
AIME Polynomials and Functions 1
Factor completely: 2x^3 - 8x.