which statement describes the graph of this polynomial function?\nf(x)=x^{5}-6x^{4}+9x^{3}\nthe graph…

which statement describes the graph of this polynomial function?\nf(x)=x^{5}-6x^{4}+9x^{3}\nthe graph crosses the x - axis at x = 0 and touches the x - axis at x = 3.\nthe graph touches the x - axis at x = 0 and crosses the x - axis at x = 3.\nthe graph crosses the x - axis at x = 0 and touches the x - axis at x = - 3.\nthe graph touches the x - axis at x = 0 and crosses the x - axis at x = - 3.

which statement describes the graph of this polynomial function?\nf(x)=x^{5}-6x^{4}+9x^{3}\nthe graph crosses the x - axis at x = 0 and touches the x - axis at x = 3.\nthe graph touches the x - axis at x = 0 and crosses the x - axis at x = 3.\nthe graph crosses the x - axis at x = 0 and touches the x - axis at x = - 3.\nthe graph touches the x - axis at x = 0 and crosses the x - axis at x = - 3.

Answer

Explanation:

Step1: Factor the polynomial

$f(x)=x^{5}-6x^{4}+9x^{3}=x^{3}(x^{2}-6x + 9)=x^{3}(x - 3)^{2}$

Step2: Analyze the roots and their multiplicities

For the root $x = 0$, the multiplicity of the factor $x^{3}$ is 3 (an odd - number). When the multiplicity of a root is odd, the graph of the polynomial crosses the x - axis at that root. For the root $x=3$, the multiplicity of the factor $(x - 3)^{2}$ is 2 (an even - number). When the multiplicity of a root is even, the graph of the polynomial touches the x - axis at that root.

Answer:

The graph crosses the x - axis at $x = 0$ and touches the x - axis at $x = 3$.