Polynomials & Rational Functions Flashcards

If α and β are the roots of x² + px + q = 0, then α² + β² equals:

Answer: p² - 2q

α + β = -p and αβ = q, so α² + β² = (α + β)² - 2αβ = (-p)² - 2q = p² - 2q.

The rational function f(x) = (x³ - x)/(x² - 1) simplifies to (for x ≠ ±1):

Answer: x

x³ - x = x(x² - 1), so (x³ - x)/(x² - 1) = x(x² - 1)/(x² - 1) = x.

By the Rational Root Theorem, which of the following is a root of p(x) = 2x³ - 3x² - 11x + 6?

Answer: 3

Possible rational roots are ±{1, 2, 3, 6}/±{1, 2}; testing x = 3: 2(27) - 3(9) - 11(3) + 6 = 54 - 27 - 33 + 6 = 0.

Given that x = 3 is a root of p(x) = 2x³ - 3x² - 11x + 6, the complete factorization is:

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

Dividing by (x - 3) via synthetic division gives 2x² + 3x - 2, which factors as (2x - 1)(x + 2).

The polynomial x⁴ + 4 factors over the integers as (Sophie Germain Identity):

Answer: (x² + 2x + 2)(x² - 2x + 2)

x⁴ + 4 = (x² + 2)² - (2x)² = (x² + 2x + 2)(x² - 2x + 2) using the difference of squares.

If a degree-4 polynomial with real coefficients has roots 2, -1, and 3 + i, what must the fourth root be?

Answer: 3 - i

Complex roots of polynomials with real coefficients come in conjugate pairs, so if 3 + i is a root, then 3 - i must also be a root.

The number of sign changes in p(x) = x⁵ - 3x⁴ + x³ + 2x² - x + 1 indicates, by Descartes' Rule of Signs, at most how many positive real roots?

Answer: 4

The signs are +, -, +, +, -, + giving sign changes at positions 1→2, 2→3, 4→5, 5→6, for 4 sign changes, so at most 4 positive real roots.