AP Pre-Calculus
AP PRE CALCULUS - QUESTION 1
Work with Desmos. Desmos is a great resource to AP Pre Calculus.
Work with Desmos
Desmos is a great resource to AP Pre Calculus
A polynomial function p is given by p(x) = -x(x - 4)(x + 2). What are all intervals on which p(x) ≥ 0?
(A) [-2, 4]
(B) [-2, 0] ∪ [4, ∞)
(C) (-∞, -4] ∪ [0, 2]
(D) (-∞, -2] ∪ [0, 4]
Step-by-step solution (AP Precalculus style):
First, find the zeros (where p(x) = 0): x = -2, 0, 4
These points divide the number line into intervals: (-∞, -2), (-2, 0), (0, 4), (4, ∞)
Now test the sign of p(x) in each interval:
- For x < -2 (try x = -3): positive
- For -2 < x < 0 (try x = -1): negative
- For 0 < x < 4 (try x = 1): positive
- For x > 4 (try x = 5): negative
Since we want p(x) ≥ 0, include where the function is positive AND the zeros:
Final answer: (-∞, -2] ∪ [0, 4]
Correct choice: D
AP Precalculus tip: For factored polynomials, always analyze sign changes across zeros. Odd multiplicity roots change sign, and the leading coefficient controls end behavior.
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