Question

In Problems 17, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor $f$over the real numbers.

$f\left(x\right)=4x^3+4x^2-7x+2$

Short answer

The real zero of the given polynomial are $-2,\frac{1}{2},\frac{1}{2}$.

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

Step-by-step solution

Given information

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

Determine the number of positive real zeros of fx using Descartes' Rule of Signs.

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

Determine the total number of sign changes.

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

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

Determine the number of negative real zeros of f-x using Descartes' Rule of Signs.

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.

$-4x^3⟶+4x^2$

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

Determine the possible factor of the given function.

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

$p=\pm 1,\pm 2$

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

$q=\pm 1,\pm 2,\pm 4$

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

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

Determine the factor of given function

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

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

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

Write the conclusion

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

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