which statement describes the graph of $f(x)=4x^{7}+40x^{6}+100x^{5}$?\nthe graph crosses the x - axis at $x…

which statement describes the graph of $f(x)=4x^{7}+40x^{6}+100x^{5}$?\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x = 5$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x = 5$.\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x=-5$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x=-5$.

which statement describes the graph of $f(x)=4x^{7}+40x^{6}+100x^{5}$?\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x = 5$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x = 5$.\nthe graph crosses the x - axis at $x = 0$ and touches the x - axis at $x=-5$.\nthe graph touches the x - axis at $x = 0$ and crosses the x - axis at $x=-5$.

Answer

Explanation:

Step1: Factor the polynomial

First, factor out the greatest - common factor from (f(x)=4x^{7}+40x^{6}+100x^{5}). The GCF is (4x^{5}), so (f(x)=4x^{5}(x^{2} + 10x + 25)). Then factor the quadratic (x^{2}+10x + 25=(x + 5)^{2}). So (f(x)=4x^{5}(x + 5)^{2}).

Step2: Find the roots and their multiplicities

Set (f(x)=0). Then (4x^{5}(x + 5)^{2}=0). Using the zero - product property, (x^{5}=0) gives (x = 0) with multiplicity (5), and ((x + 5)^{2}=0) gives (x=-5) with multiplicity (2).

Step3: Determine the behavior at the roots

If the multiplicity of a root (r) of a polynomial (y = f(x)) is odd, the graph of the polynomial crosses the (x) - axis at (x = r). If the multiplicity is even, the graph of the polynomial touches the (x) - axis at (x = r). Since the multiplicity of (x = 0) is (5) (odd), the graph crosses the (x) - axis at (x = 0). Since the multiplicity of (x=-5) is (2) (even), the graph touches the (x) - axis at (x=-5).

Answer:

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