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.

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."