Polynomials & Rational Functions Flashcards

By Vieta's formulas, the product of the roots of 3x⁴ - 2x³ + x² - 5x + 6 = 0 is:

Answer: 2

By Vieta's formulas, the product of all roots equals the constant term divided by the leading coefficient: 6/3 = 2.

What is the remainder when p(x) = x¹⁰⁰ - 1 is divided by (x - 1)?

Answer: 0

By the Remainder Theorem, the remainder is p(1) = 1¹⁰⁰ - 1 = 0.

The polynomial x⁴ + x² + 1 has how many real roots?

Answer: 0

Since x⁴ ≥ 0 and x² ≥ 0, we have x⁴ + x² + 1 ≥ 1 > 0 for all real x, so there are no real roots.

If p(x) is a polynomial with p(2) = 5 and p(3) = 7, the remainder when p(x) is divided by (x-2)(x-3) is:

Answer: 2x + 1

The remainder R(x) = ax + b satisfies R(2) = 2a + b = 5 and R(3) = 3a + b = 7, giving a = 2, b = 1, so R(x) = 2x + 1.

The polynomial x³ - 3x² + 3x - 1 = 0 has how many distinct positive real roots?

Answer: 1

x³ - 3x² + 3x - 1 = (x - 1)³, so x = 1 is the only root (with multiplicity 3).

If f(x) = (x² - 4)/(x - 2) for x ≠ 2, then f(x) simplifies to:

Answer: x + 2

x² - 4 = (x - 2)(x + 2), so (x² - 4)/(x - 2) = x + 2 for x ≠ 2.

For the polynomial p(x) = xⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ with roots r₁, r₂, …, rₙ, by Vieta's formulas the sum r₁ + r₂ + … + rₙ equals:

Answer: -aₙ₋₁

For a monic polynomial, Vieta's formulas give the sum of roots as the negative of the coefficient of xⁿ⁻¹.