# How do you find all the zeros of  f(x)=2x^3-13x^2+22x-8?

Apr 1, 2016

Use the Rational Factor Theorem to identify rational zeros.

#### Explanation:

According to the Rational Factor Theorem
if $2 {x}^{3} - 13 {x}^{2} + 22 x - 8$ has rational zeros
they must be in the set
$\textcolor{w h i t e}{\text{XXX")+-("factors of 8")/("factors of 2}} = \pm \frac{\left\{1 , 2 , 4 , 8\right\}}{\left\{1 , 2\right\}}$
Evaluating the equation for each possibility:

identifies the zeros as $\left\{\textcolor{red}{\frac{1}{2}} , \textcolor{b l u e}{2} , \textcolor{b r o w n}{4}\right\}$
which implies the factoring:
$\textcolor{w h i t e}{\text{XXX}} \left(\textcolor{red}{2 x - 1}\right) \left(\textcolor{b l u e}{x - 2}\right) \left(\textcolor{b r o w n}{x - 4}\right)$