DSAT Advanced Math 3 — Questions and Answers
Question 1: A polynomial p(x) satisfies p(2) = 0. Which of the following must be a factor of p(x)?
- x + 2
- x - 2 (Correct answer)
- x² - 2
- 2x - 1
Correct answer: x - 2
By the Factor Theorem, if p(2) = 0, then (x - 2) is a factor of p(x).
Question 2: What is the sum of all solutions to x² - 5x + 6 = 0?
- 2
- 3
- 5 (Correct answer)
- 6
Correct answer: 5
By Vieta's formulas, the sum of roots = -b/a = 5/1 = 5.
Question 3: Which of the following is equivalent to (x + 3)² - 4?
- x² + 6x + 5 (Correct answer)
- x² + 6x + 13
- x² + 9x + 5
- x² + 3x + 5
Correct answer: x² + 6x + 5
Expand (x+3)² - 4 = x² + 6x + 9 - 4 = x² + 6x + 5.
Question 4: For what value of k does the equation x² + kx + 16 = 0 have exactly one real solution?
- 4
- 6
- 8 (Correct answer)
- 16
Correct answer: 8
For one solution the discriminant = 0: k² - 4(16) = 0 → k² = 64 → k = 8.
Question 5: The graph of y = a(x - h)² + k passes through (0, 5) with vertex at (2, 1). What is the value of a?
- 1 (Correct answer)
- 4/3
- 1/2
- 2
Correct answer: 1
Substituting (0,5): 5 = a(0-2)² + 1 = 4a + 1, so 4a = 4 and a = 1.
Question 6: Which expression is equivalent to (x² + 2x - 8) / (x + 4)?
- x - 2 (Correct answer)
- x + 2
- x - 4
- x + 4
Correct answer: x - 2
Factor numerator as (x+4)(x-2), then cancel (x+4) to get x-2.
Question 7: If f(x) = x³ - x and g(x) = x + 2, what is f(g(1))?
- 6
- 18
- 24 (Correct answer)
- 12
Correct answer: 24
g(1) = 3, then f(3) = 27 - 3 = 24.
DSAT Advanced Math 3
A polynomial p(x) satisfies p(2) = 0.
Which of the following must be a factor of p(x)?