PRE CALCULUS Pre-Calculus Polynomial Functions 3 — Questions and Answers
Question 1: Using synthetic division, what is the quotient when x³ + 2x² - 5x - 6 is divided by (x - 2)?
- x² + 4x + 3 (Correct answer)
- x² - 4x + 3
- x² + 4x - 3
- x² - 4x - 3
Correct answer: x² + 4x + 3
Synthetic division with c = 2 yields coefficients 1, 4, 3, remainder 0, giving x² + 4x + 3.
Question 2: According to the Rational Zero Theorem, which is a possible rational zero of f(x) = 2x³ - 3x + 6?
- ±4
- ±3/2 (Correct answer)
- ±5
- ±7
Correct answer: ±3/2
Possible rational zeros are factors of the constant (6) over factors of the leading coefficient (2), giving ±3/2 among others.
Question 3: How many positive real zeros can f(x) = x⁴ - x³ + x² - x + 1 have according to Descartes' Rule?
- 0 or 2 or 4 (Correct answer)
- 1 or 3
- 0 only
- 4 only
Correct answer: 0 or 2 or 4
There are 4 sign changes, so Descartes' Rule gives 4, 2, or 0 positive real zeros.
Question 4: If f(x) has degree 6 with real coefficients and 2 + 3i is a zero, what other zero must exist?
- 2 - 3i (Correct answer)
- -2 + 3i
- 3 - 2i
- -2 - 3i
Correct answer: 2 - 3i
Complex zeros of polynomials with real coefficients come in conjugate pairs, so 2 - 3i must also be a zero.
Question 5: What is the remainder when f(x) = x⁴ - 3x² + 2 is divided by (x + 1)?
- 0 (Correct answer)
- 2
- 4
- -2
Correct answer: 0
By the Remainder Theorem, f(-1) = 1 - 3 + 2 = 0.
Question 6: Which graph feature indicates a zero of multiplicity 2?
- The graph crosses the x-axis steeply
- The graph bounces off the x-axis without crossing (Correct answer)
- The graph has a vertical asymptote
- The graph crosses the y-axis twice
Correct answer: The graph bounces off the x-axis without crossing
A zero with even multiplicity causes the graph to touch the x-axis and turn back, not cross it.
Question 7: For f(x) = (x - 1)²(x + 3), what are the zeros and their multiplicities?
- x = 1 (mult. 1), x = -3 (mult. 2)
- x = 1 (mult. 2), x = -3 (mult. 1) (Correct answer)
- x = -1 (mult. 2), x = 3 (mult. 1)
- x = 1 (mult. 2), x = 3 (mult. 1)
Correct answer: x = 1 (mult. 2), x = -3 (mult. 1)
The factor (x - 1)² gives zero x = 1 with multiplicity 2, and (x + 3) gives x = -3 with multiplicity 1.
PRE CALCULUS Pre-Calculus Polynomial Functions 3
Using synthetic division, what is the quotient when x³ + 2x² - 5x - 6 is divided by (x - 2)?