which statement describes the graph of $f(x)=-x^{4}+3x^{3}+10x^{2}$?\nthe graph crosses the x - axis at $x =…

which statement describes the graph of $f(x)=-x^{4}+3x^{3}+10x^{2}$?\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x = 5$ and $x=-2$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x = 5$ and $x=-2$.\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x=-5$ and $x = 2$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x=-5$ and $x = 2$.

which statement describes the graph of $f(x)=-x^{4}+3x^{3}+10x^{2}$?\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x = 5$ and $x=-2$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x = 5$ and $x=-2$.\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x=-5$ and $x = 2$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x=-5$ and $x = 2$.

Answer

Answer:

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

Explanation:

Step1: Factor the polynomial

First, factor (f(x)=-x^{4}+3x^{3}+10x^{2}) as (f(x)=-x^{2}(x^{2}-3x - 10)). Then factor (x^{2}-3x - 10=(x - 5)(x+2)), so (f(x)=-x^{2}(x - 5)(x + 2)).

Step2: Analyze the roots

The roots of the function are (x = 0), (x = 5) and (x=-2). For a root (x = a) of a polynomial (y = P(x)), if the factor ((x - a)) has an even exponent, the graph touches the x - axis at (x = a), and if the factor ((x - a)) has an odd exponent, the graph crosses the x - axis at (x = a). The factor ((x-0)) has exponent 2 (even), so the graph touches the x - axis at (x = 0), and the factors ((x - 5)) and ((x + 2)) have exponent 1 (odd), so the graph crosses the x - axis at (x = 5) and (x=-2).