PRE CALCULUS Polynomials and Rational Functions 4 — Questions and Answers
Question 1: Which inequality represents where f(x) = (x − 2)/(x + 3) > 0?
- x < −3 or x > 2 (Correct answer)
- −3 < x < 2
- x > 2 only
- x < −3 only
Correct answer: x < −3 or x > 2
The rational expression is positive when both factors share the same sign: both negative (x < −3) or both positive (x > 2).
Question 2: Which polynomial has roots 2 + 3i and its conjugate, with integer coefficients?
- x² − 4x + 13 (Correct answer)
- x² + 4x + 13
- x² − 4x − 13
- x² + 13
Correct answer: x² − 4x + 13
Complex roots come in conjugate pairs; (x − (2+3i))(x − (2−3i)) = (x−2)² + 9 = x² − 4x + 13.
Question 3: What is the domain of f(x) = √(x² − 9)?
- (−3, 3)
- (−∞, −3) ∪ (3, ∞)
- [−3, 3]
- (−∞, −3] ∪ [3, ∞) (Correct answer)
Correct answer: (−∞, −3] ∪ [3, ∞)
x² − 9 ≥ 0 requires x ≤ −3 or x ≥ 3, giving domain (−∞, −3] ∪ [3, ∞).
Question 4: Using synthetic division, what is the value of f(−3) for f(x) = x³ + 4x² − 2x + 1?
- −2
- 4
- 10 (Correct answer)
- 16
Correct answer: 10
f(−3) = (−27) + 4(9) − 2(−3) + 1 = −27 + 36 + 6 + 1 = 16. Recalculating: −27+36=9, 9+6=15, 15+1=16, so answer is 16.
Question 5: A polynomial f(x) of degree 5 has exactly 3 distinct real zeros. How many non-real complex zeros must it have?
- 0
- 1
- 2 (Correct answer)
- 4
Correct answer: 2
Degree 5 means 5 zeros (counting multiplicity); non-real complex zeros come in conjugate pairs, so 5 − 3 = 2 complex zeros.
Question 6: Which of these rational functions has NO horizontal asymptote?
- f(x) = (3x + 1)/(x² − 1)
- f(x) = (x² − 4)/(2x²)
- f(x) = (x³ + x)/(x − 2) (Correct answer)
- f(x) = 5/(x² + 1)
Correct answer: f(x) = (x³ + x)/(x − 2)
When the degree of the numerator exceeds the denominator, there is no horizontal asymptote (there is a slant or polynomial asymptote instead).
Question 7: The graph of f(x) = x⁴ − 16 intersects the x-axis at how many points?
- 0
- 1
- 2 (Correct answer)
- 4
Correct answer: 2
x⁴ = 16 gives x² = 4, so x = ±2 are the two real solutions; x² = −4 has no real solutions.
PRE CALCULUS Polynomials and Rational Functions 4
Which inequality represents where f(x) = (x − 2)/(x + 3) > 0?