91Ó°ÊÓ

The given function is $f\left(x\right)=4x^3+4x^2-7x+2$.

Identify any change in signs from one term to the next term in the given function.

Determine the total number of sign changes.

$$\begin{gathered} +4x^2→-7x \\ -7x→+2 \end{gathered}$$

Since $f\left(x\right)$has 2 sign changes, there are two positive real zeros or no positive real zeros.

First, evaluate $f\left(-x\right)$.

$\begin{array}{rcl}f(-x)& =& 4(-x{)}^3+4(-x{)}^2-7(-x)+2\\ & =& -4x^3+4x^2+7x+2\end{array}$

Identify any change in signs from one term to the next term in the given function.

Determine the total number of sign changes.

$-4x^3âŸ\P +4x^2$

Since $f\left(-x\right)$has 1 sign change, there is one negative real zero.

List all the possible factors $p$of $a_0$.

$p=Â\pm 1,Â\pm 2$

List all the possible factors $q$of $a_3$.

$q=Â\pm 1,Â\pm 2,Â\pm 4$

Identify all possible rational zeros $\frac{p}{q}$.

$\frac{p}{q}=Â\pm 1,Â\pm 2,Â\pm \frac{1}{2},Â\pm \frac{1}{4}$

Perform long division to check if$x+2$ is a factor of $f\left(x\right)$.

Now, the function is written as $f\left(x\right)=\left(x+2\right)\left(4x^2-4x+1\right)$

$$\begin{gathered} f\left(x\right)=\left(x+2\right)\left(4x^2-4x+1\right) \\ =(x+2)(2x-1)(2x-1) \\ =4\left(x+2\right)\left(x-\frac{1}{2}\right)\left(x-\frac{1}{2}\right) \end{gathered}$$

The zeros of $f\left(x\right)$are $-2$with multiplicity 1 , and $\frac{1}{2}$with multiplicity 2 .

$f\left(x\right)=4(x+2)\left(x-\frac{1}{2}\right)\left(x-\frac{1}{2}\right)$