PRE CALCULUS Polynomials and Rational Functions 5 — Questions and Answers
Question 1: If f(x) = 2x³ − 7x² + 2x + 3, and x = 3 is a zero, which factored form is correct?
- (x − 3)(2x² − x − 1) (Correct answer)
- (x − 3)(2x² + x + 1)
- (x + 3)(2x² − x − 1)
- (x − 3)(2x² − 5x)
Correct answer: (x − 3)(2x² − x − 1)
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?
- −5
- −4
- 4 (Correct answer)
- 12
Correct answer: 4
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?
- The function approaches 0
- The function has a hole
- The function goes to ±∞ (Correct answer)
- The function equals 1/4
Correct answer: The function goes to ±∞
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.
- −6
- 6 (Correct answer)
- 11
- −11
Correct answer: 6
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?
- The graph touches the x-axis and bounces back
- The graph crosses through the x-axis (Correct answer)
- The graph has a vertical asymptote there
- The graph is undefined there
Correct answer: The graph crosses through the x-axis
At a zero of odd multiplicity, the polynomial changes sign so the graph crosses through the x-axis.
Question 6: Solve the inequality (x² − 4)/(x + 1) ≥ 0. Which interval is part of the solution?
- (−2, −1)
- (−1, 2)
- x = −2
- (−∞, −2] (Correct answer)
Correct answer: (−∞, −2]
Testing intervals around x = −2, −1, and 2 shows the expression is non-negative on (−∞, −2] and [2, ∞), excluding x = −1.
Question 7: How many turning points can a degree-6 polynomial have at most?
- 4
- 5 (Correct answer)
- 6
- 7
Correct answer: 5
A polynomial of degree n has at most n − 1 turning points, so a degree-6 polynomial has at most 5.
PRE CALCULUS Polynomials and Rational Functions 5
If f(x) = 2x³ − 7x² + 2x + 3, and x = 3 is a zero, which factored form is correct?