Question

Find the zeros of the function f(x) = x3 − 2x2 − 19x + 20, and describe the behavior of the graph at each zero.
Responses

A. The graph touches the x-axis at 4, and it crosses the x-axis at −1 and −5.
A. The graph touches the x -axis at 4, and it crosses the x -axis at −1 and −5.

B. The graph touches the x-axis at 4, −5, and 1.
B. The graph touches the x -axis at 4, −5, and 1.

C. The graph crosses the x-axis at 5, 1, and −4.
C. The graph crosses the x -axis at 5, 1, and −4.

D. The graph touches the x-axis at −4, and it crosses the x-axis at 1 and 5.

Answer 1

f(x) = x^3 − 2x^2 − 19x + 20. Test x = 1: f(1)=0, so divide by (x−1) to get x^2 − x − 20 = (x−5)(x+4). Zeros: x = 1, 5, −4, each with multiplicity 1. Since multiplicities are odd (1), the graph crosses the x-axis at all three zeros. Choice C.