Which statement describes the graph of f (x) = - x4 + 3x3 + 10x2?

The graph crosses the x axis at x = 0 and touches the x axis at x = 5 and x = - 2.

The graph touches the x axis at x = 0 and crosses the x axis at x = 5 and x = - 2.

The graph crosses the x axis at x = 0 and touches the x axis at x = - 5 and x = 2.

The graph touches the x axis at x = 0 and crosses the x axis at x = - 5 and x = 2.

Question

Mathematics

Shayna Simmons

8 September, 18:22

Which statement describes the graph of f (x) = - x4 + 3x3 + 10x2?

The graph crosses the x axis at x = 0 and touches the x axis at x = 5 and x = - 2.

The graph touches the x axis at x = 0 and crosses the x axis at x = 5 and x = - 2.

The graph crosses the x axis at x = 0 and touches the x axis at x = - 5 and x = 2.

The graph touches the x axis at x = 0 and crosses the x axis at x = - 5 and x = 2.

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Answers (1)

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Kristian Arnold · Answer 1

f (x) = - x^4 + 3x^3 + 10x^2 = - x^2 (x^2 - 3x - 10) = - x^2 (x^2 + 2x - 5x - 10) = - x^2[x (x + 2) - 5 (x + 2) ] = - x^2 (x - 5) (x + 2) Thus f (x) = - x^4 + 3x^3 + 10x^2 has roots of 0 with multiplicity of 2, 5 with multiplicity of 1 and - 2 with multiplicity of 1. Therefore, the graph of f (x) = - x^4 + 3x^3 + 10x^2 touches the x-axis at x = 0 and crosses the x-axis at x = 5 and x = - 2. The correct answer to the question "Which statement describes the graph of f (x) = - x^4 + 3x^3 + 10x^2?" is "The graph touches the x axis at x = 0 and crosses the x axis at x = 5 and x = - 2."