AMC 12 Polynomials & Rational Functions 2

Question 1

If the polynomial x³ + ax² + bx + c has roots 1, 2, and 3, what is a + b + c?

Accepted Answer

-1

Answer

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?

Accepted Answer

x² + 2x + 4

Answer

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:

Accepted Answer

y = 2

Answer

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:

Accepted Answer

19

Answer

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?

Accepted Answer

2

Answer

The denominator is zero at x = ±2; since x² - 9 ≠ 0 at these points, both give vertical asymptotes, for a total of 2.

AMC 12 - - American Mathematics Contest 12 Practice Test