AMC 12 Polynomials & Rational Functions 2 — Questions and Answers
Question 1: If the polynomial x³ + ax² + bx + c has roots 1, 2, and 3, what is a + b + c?
- -1 (Correct answer)
- 1
- 6
- -6
Correct answer: -1
By Vieta's: a = -(1+2+3) = -6, b = 1·2+1·3+2·3 = 11, c = -(1·2·3) = -6, so a + b + c = -6 + 11 - 6 = -1.
Question 2: Which of the following is a factor of x³ - 8?
- x² + 2x + 4 (Correct answer)
- x² - 2x + 4
- x + 2
- x² + 4
Correct answer: x² + 2x + 4
Using the difference of cubes: x³ - 8 = (x - 2)(x² + 2x + 4).
Question 3: For the rational function f(x) = (2x² + 3)/(x² - 1), the horizontal asymptote is:
- y = 2 (Correct answer)
- y = -3
- y = 0
- y = 3
Correct answer: y = 2
When the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients: 2/1 = 2.
Question 4: The sum of the squares of the roots of x² - 5x + 3 = 0 equals:
- 19 (Correct answer)
- 25
- 13
- 22
Correct answer: 19
If α, β are the roots, then α² + β² = (α + β)² - 2αβ = 5² - 2(3) = 25 - 6 = 19.
Question 5: How many vertical asymptotes does f(x) = (x² - 9)/(x² - 4) have?
- 2 (Correct answer)
- 0
- 1
- 4
Correct answer: 2
The denominator is zero at x = ±2; since x² - 9 ≠ 0 at these points, both give vertical asymptotes, for a total of 2.
Question 6: The polynomial p(x) = x⁴ - 5x² + 4 factors completely over the integers as:
- (x-1)(x+1)(x-2)(x+2) (Correct answer)
- (x²-1)(x-4)
- (x-2)²(x+1)²
- (x²+4)(x²-1)
Correct answer: (x-1)(x+1)(x-2)(x+2)
Substituting u = x²: u² - 5u + 4 = (u-1)(u-4) = (x²-1)(x²-4) = (x-1)(x+1)(x-2)(x+2).
Question 7: For how many distinct real values of x does f(x) = x³ - 4x = 0?
- 3 (Correct answer)
- 1
- 2
- 4
Correct answer: 3
x³ - 4x = x(x² - 4) = x(x - 2)(x + 2) = 0, giving x = 0, 2, -2 — three distinct values.
AMC 12 Polynomials & Rational Functions 2
If the polynomial x³ + ax² + bx + c has roots 1, 2, and 3, what is a + b + c?