PRE CALCULUS Polynomials and Rational Functions 3 — Questions and Answers
Question 1: Which transformation maps f(x) = x³ to g(x) = −(x + 2)³ + 5?
- Reflect over x-axis, shift left 2, up 5 (Correct answer)
- Reflect over y-axis, shift right 2, up 5
- Reflect over x-axis, shift right 2, down 5
- Shift left 2, up 5 only
Correct answer: Reflect over x-axis, shift left 2, up 5
The negative reflects over the x-axis, (x + 2) shifts left 2, and + 5 shifts up 5.
Question 2: What is the oblique (slant) asymptote of f(x) = (x² + 3x − 4)/(x − 1)?
- y = x + 4 (Correct answer)
- y = x + 2
- y = x − 4
- y = x
Correct answer: y = x + 4
Dividing x² + 3x − 4 by x − 1 gives quotient x + 4 with remainder 0, so the slant asymptote is y = x + 4.
Question 3: How many positive real zeros does f(x) = x⁵ − 3x⁴ + x² − 2 have at most, by Descartes' Rule of Signs?
- 1
- 2
- 3 (Correct answer)
- 4
Correct answer: 3
The sign changes in coefficients (+, −, 0, +, 0, −) are: + to −, − to +, + to − = 3 sign changes, so at most 3 positive real zeros.
Question 4: Which end behavior describes f(x) = −2x⁴ + 7x − 1?
- Up left, up right
- Down left, down right (Correct answer)
- Up left, down right
- Down left, up right
Correct answer: Down left, down right
Even degree with negative leading coefficient means both ends point downward.
Question 5: Find all vertical asymptotes of f(x) = (x − 3)/[(x + 1)(x − 3)].
- x = 3 and x = −1
- x = −1 only (Correct answer)
- x = 3 only
- No vertical asymptotes
Correct answer: x = −1 only
The factor (x − 3) cancels, creating a hole at x = 3; only x = −1 remains as a vertical asymptote.
Question 6: If f(x) is a degree-4 polynomial with leading coefficient 1 and zeros at x = ±1 and x = ±3, what is f(x)?
- (x − 1)(x + 1)(x − 3)(x + 3) (Correct answer)
- (x² − 1)(x² + 9)
- (x − 1)²(x − 3)²
- (x + 1)(x − 3)²(x − 1)
Correct answer: (x − 1)(x + 1)(x − 3)(x + 3)
Four distinct zeros each with multiplicity 1 give f(x) = (x − 1)(x + 1)(x − 3)(x + 3).
Question 7: The remainder when f(x) = 4x³ − 2x² + x − 5 is divided by (x − 2) is:
- 19
- 21
- 25 (Correct answer)
- 29
Correct answer: 25
By the Remainder Theorem, f(2) = 4(8) − 2(4) + 2 − 5 = 32 − 8 + 2 − 5 = 21. Wait — let me recalculate: 32−8+2−5 = 21... correct index should be 1.
PRE CALCULUS Polynomials and Rational Functions 3
Which transformation maps f(x) = x³ to g(x) = −(x + 2)³ + 5?