PRE CALCULUS Polynomials and Rational Functions 5

Question 1

If f(x) = 2x³ − 7x² + 2x + 3, and x = 3 is a zero, which factored form is correct?

Accepted Answer

(x − 3)(2x² − x − 1)

Answer

Dividing by (x − 3) via synthetic division yields 2x² − x − 1, giving f(x) = (x − 3)(2x² − x − 1).

Question 2

What is the sum of all zeros of f(x) = 3x⁴ − 12x³ + 5x − 1?

Accepted Answer

4

Answer

By Vieta's formulas, the sum of zeros equals −(coefficient of x³)/(leading coefficient) = −(−12)/3 = 4.

Question 3

Which describes the behavior of f(x) = 1/(x − 4) near x = 4?

Accepted Answer

The function goes to ±∞

Answer

As x → 4, the denominator approaches 0 and the function grows without bound (vertical asymptote at x = 4).

Question 4

Find the product of all zeros of f(x) = x³ − 6x² + 11x − 6.

Accepted Answer

6

Answer

By Vieta's formulas for a monic cubic x³ + bx² + cx + d, the product of zeros = −d/1 = −(−6) = 6.

Question 5

Which of the following correctly describes a zero of odd multiplicity?

Accepted Answer

The graph crosses through the x-axis

Answer

At a zero of odd multiplicity, the polynomial changes sign so the graph crosses through the x-axis.

PRE CALCULUS Practice Test