PRE CALCULUS Polynomials and Rational Functions 2 — Questions and Answers
Question 1: Which of the following is a factor of f(x) = x³ − 6x² + 11x − 6?
- (x − 4)
- (x − 3) (Correct answer)
- (x + 2)
- (x + 6)
Correct answer: (x − 3)
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)?
- 1
- 2
- 3 (Correct answer)
- 4
Correct answer: 3
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?
- (x + 2)
- (x − 2) (Correct answer)
- (x² − 4)
- (x − 4)
Correct answer: (x − 2)
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?
- x³ + x² + x + 1 (Correct answer)
- x³ − x² + x − 1
- x³ + 1
- x³ − 1
Correct answer: x³ + x² + x + 1
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)?
- f(x) = (x − 5)(x + 1)
- f(x) = (x − 5)²(x + 1)
- f(x) = (x − 5)(x + 1)² (Correct answer)
- f(x) = (x − 5)²(x + 1)²
Correct answer: f(x) = (x − 5)(x + 1)²
Crossing requires odd multiplicity at x = 5, and touching requires even multiplicity at x = −1, so (x − 5)(x + 1)².
Question 6: For the rational function f(x) = (3x² + 1)/(x² − 4), what is the horizontal asymptote?
- y = 0
- y = 1
- y = 3 (Correct answer)
- There is none
Correct answer: y = 3
When the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients: 3/1 = 3.
Question 7: Using the Rational Root Theorem, which is a possible rational root of 2x³ − 5x² + 3?
- ±4
- ±6
- ±3/2 (Correct answer)
- ±5
Correct answer: ±3/2
Possible rational roots are factors of the constant (3) over factors of the leading coefficient (2): ±1, ±3, ±1/2, ±3/2.
PRE CALCULUS Polynomials and Rational Functions 2
Which of the following is a factor of f(x) = x³ − 6x² + 11x − 6?