Question

Which statement describes the graph of this polynomial function?

f (x) = x Superscript 5 Baseline minus 6 x Superscript 4 Baseline + 9 x cubed
The graph crosses the x-axis at x = 0 and touches the x-axis at x = 3.
The graph touches the x-axis at x = 0 and crosses the x-axis at x = 3.
The graph crosses the x-axis at x = 0 and touches the x-axis at x = –3.
The graph touches the x-axis at x = 0 and crosses the x-axis at x = –3.

Answer 1

Factor: f(x) = x^3(x - 3)^2. The zero x = 0 has multiplicity 3 (odd) so the graph crosses there; x = 3 has multiplicity 2 (even) so the graph touches (bounces) there.

Answer: The graph crosses at x = 0 and touches at x = 3.