PRE CALCULUS Polynomials and Rational Functions 2

Question 1

Which of the following is a factor of f(x) = x³ − 6x² + 11x − 6?

Accepted Answer

(x − 3)

Answer

f(3) = 27 − 54 + 33 − 6 = 0, so (x − 3) is a factor by the Factor Theorem.

Question 2

What is the multiplicity of the root x = −2 in f(x) = (x + 2)³(x − 1)?

Accepted Answer

3

Answer

The factor (x + 2) appears raised to the 3rd power, so x = −2 has multiplicity 3.

Question 3

A rational function has a hole at x = 2. Which factor must appear in both the numerator and denominator?

Accepted Answer

(x − 2)

Answer

A hole occurs when a common factor (x − 2) cancels from both numerator and denominator.

Question 4

What is the quotient when x⁴ − 1 is divided by x − 1?

Accepted Answer

x³ + x² + x + 1

Answer

Synthetic or polynomial division of x⁴ − 1 by x − 1 gives x³ + x² + x + 1.

Question 5

The graph of a polynomial crosses the x-axis at x = 5 and touches (but does not cross) at x = −1. Which could be f(x)?

Accepted Answer

f(x) = (x − 5)(x + 1)²

Answer

Crossing requires odd multiplicity at x = 5, and touching requires even multiplicity at x = −1, so (x − 5)(x + 1)².

PRE CALCULUS Practice Test