which statement describes the graph of f(x)=-x^4 + 3x^3 + 10x^2? the graph crosses the x - axis at x = 0 and…

which statement describes the graph of f(x)=-x^4 + 3x^3 + 10x^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. 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.

which statement describes the graph of f(x)=-x^4 + 3x^3 + 10x^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. 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.

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}). Factor out (-x^{2}) to get (f(x)=-x^{2}(x^{2} - 3x - 10)). Then factor the quadratic (x^{2}-3x - 10=(x - 5)(x+2)). So (f(x)=-x^{2}(x - 5)(x + 2)).

Step2: Analyze the roots and their multiplicities

The roots of the function are found by setting (f(x)=0). We have (x = 0) with multiplicity 2, (x = 5) with multiplicity 1, and (x=-2) with multiplicity 1. If the multiplicity of a root is even, the graph touches the x - axis at that point. If the multiplicity is odd, the graph crosses the x - axis at that point. Since the multiplicity of (x = 0) is 2 (even), the graph touches the x - axis at (x = 0). Since the multiplicities of (x = 5) and (x=-2) are 1 (odd), the graph crosses the x - axis at (x = 5) and (x=-2).