the graph of the polynomial ( f(x) ) is given below.\nif ( f(x) ) has degree 4, find the factored…

the graph of the polynomial ( f(x) ) is given below.\nif ( f(x) ) has degree 4, find the factored equation\nfor ( f(x) ).\n\nselect the correct answer below:\n\n( \bigcirc f(x)=-\frac{1}{8}(x - 2)(x - 1)(x + 2)^{2} )\n( \bigcirc f(x)=\frac{1}{8}(x - 2)(x - 1)(x + 2)^{2} )\n( \bigcirc f(x)=-\frac{1}{16}(x - 2)(x - 1)(x + 2)^{2} )\n( \bigcirc f(x)=-\frac{1}{8}(x - 2)^{2}(x + 1)(x + 2) )\n( \bigcirc f(x)=\frac{1}{16}(x - 2)^{2}(x + 1)(x + 2) )\n( \bigcirc f(x)=\frac{1}{8}(x - 2)^{2}(x + 1)(x + 2) )

the graph of the polynomial ( f(x) ) is given below.\nif ( f(x) ) has degree 4, find the factored equation\nfor ( f(x) ).\n\nselect the correct answer below:\n\n( \bigcirc f(x)=-\frac{1}{8}(x - 2)(x - 1)(x + 2)^{2} )\n( \bigcirc f(x)=\frac{1}{8}(x - 2)(x - 1)(x + 2)^{2} )\n( \bigcirc f(x)=-\frac{1}{16}(x - 2)(x - 1)(x + 2)^{2} )\n( \bigcirc f(x)=-\frac{1}{8}(x - 2)^{2}(x + 1)(x + 2) )\n( \bigcirc f(x)=\frac{1}{16}(x - 2)^{2}(x + 1)(x + 2) )\n( \bigcirc f(x)=\frac{1}{8}(x - 2)^{2}(x + 1)(x + 2) )

Answer

Answer:

A. ( f(x) = -\frac{1}{8}(x - 2)(x - 1)(x + 2)^2 )

Explanation:

Step1: Determine the roots

The roots of the polynomial are (x = - 2) (with multiplicity (2) since the graph touches the (x)-axis at (x=-2)), (x = 1), and (x = 2). So the polynomial can be written in the form (f(x)=a(x + 2)^2(x - 1)(x - 2)).

Step2: Find the leading coefficient (a)

Use the (y)-intercept ((0,-1)). Substitute (x = 0) and (y=-1) into (f(x)=a(x + 2)^2(x - 1)(x - 2)): [ \begin{align*} -1&=a(0 + 2)^2(0 - 1)(0 - 2)\ -1&=a\times4\times(-1)\times(-2)\ -1&=8a\ a&=-\frac{1}{8} \end{align*} ]

So the factored equation is (f(x)=-\frac{1}{8}(x + 2)^2(x - 1)(x - 2)), which is equivalent to (f(x)=-\frac{1}{8}(x - 2)(x - 1)(x + 2)^2).