which statement describes the graph of $f(x)=-4x^{3}-28x^{2}-32x + 64$?\nthe graph crosses the x - axis at…

which statement describes the graph of $f(x)=-4x^{3}-28x^{2}-32x + 64$?\nthe graph crosses the x - axis at $x = 4$ and touches the x - axis at $x=-1$.\nthe graph touches the x - axis at $x = 4$ and crosses the x - axis at $x=-1$.\nthe graph crosses the x - axis at $x=-4$ and touches the x - axis at $x = 1$.\nthe graph touches the x - axis at $x=-4$ and crosses the x - axis at $x = 1$.

which statement describes the graph of $f(x)=-4x^{3}-28x^{2}-32x + 64$?\nthe graph crosses the x - axis at $x = 4$ and touches the x - axis at $x=-1$.\nthe graph touches the x - axis at $x = 4$ and crosses the x - axis at $x=-1$.\nthe graph crosses the x - axis at $x=-4$ and touches the x - axis at $x = 1$.\nthe graph touches the x - axis at $x=-4$ and crosses the x - axis at $x = 1$.

Answer

Answer:

The graph crosses the x - axis at (x=-4) and touches the x - axis at (x = 1).

Explanation:

Step1: Factor the polynomial

First, factor out (-4) from (f(x)=-4x^{3}-28x^{2}-32x + 64) to get (f(x)=-4(x^{3}+7x^{2}+8x - 16)). By trial - and - error, we find that (x = 1) is a root. Using synthetic division or long - division: ((x^{3}+7x^{2}+8x - 16)\div(x - 1)=x^{2}+8x + 16). So (f(x)=-4(x - 1)(x^{2}+8x + 16)=-4(x - 1)(x + 4)^{2}).

Step2: Analyze the roots

For a polynomial (y = a(x - r_{1})^{m_{1}}(x - r_{2})^{m_{2}}\cdots(x - r_{n})^{m_{n}}), if (m_{i}) is odd, the graph crosses the x - axis at (x=r_{i}), and if (m_{i}) is even, the graph touches the x - axis at (x=r_{i}). For (f(x)=-4(x - 1)(x + 4)^{2}), the root (x = 1) has multiplicity (m_1=1) (odd), so the graph crosses the x - axis at (x = 1). The root (x=-4) has multiplicity (m_2 = 2) (even), so the graph touches the x - axis at (x=-4).