Polynomials & Rational Functions Flashcards

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

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.

Which of the following is a factor of x³ - 8?

Answer: x² + 2x + 4

Using the difference of cubes: x³ - 8 = (x - 2)(x² + 2x + 4).

For the rational function f(x) = (2x² + 3)/(x² - 1), the horizontal asymptote is:

Answer: y = 2

When the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients: 2/1 = 2.

The sum of the squares of the roots of x² - 5x + 3 = 0 equals:

Answer: 19

If α, β are the roots, then α² + β² = (α + β)² - 2αβ = 5² - 2(3) = 25 - 6 = 19.

How many vertical asymptotes does f(x) = (x² - 9)/(x² - 4) have?

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.

The polynomial p(x) = x⁴ - 5x² + 4 factors completely over the integers as:

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).

For how many distinct real values of x does f(x) = x³ - 4x = 0?

Answer: 3

x³ - 4x = x(x² - 4) = x(x - 2)(x + 2) = 0, giving x = 0, 2, -2 — three distinct values.