AMC 12 Polynomials & Rational Functions

Question 1

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

Accepted Answer

2

Answer

-2

Answer

3

Answer

-3

Question 2

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

Accepted Answer

0

Answer

1

Answer

-1

Answer

100

Question 3

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

Accepted Answer

0

Answer

1

Answer

2

Answer

4

Question 4

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:

Accepted Answer

2x + 1

Answer

2x - 1

Answer

x + 3

Answer

3x - 1

Question 5

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

Accepted Answer

1

Answer

2

Answer

3

Answer

0

Question 6

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

Accepted Answer

x + 2

Answer

x - 2

Answer

x² - 4

Answer

x + 4

Question 7

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:

Accepted Answer

-aₙ₋₁

Answer

aₙ₋₁

Answer

a₀

Answer

-a₀