# How do you use the rational root theorem to find the roots of x^4 – 7x^2 + 12 = 0?

Oct 28, 2015

Evaluation of ${x}^{4} - {x}^{2} + 12$ for all factors of $12$
gives the roots $x = 2$ and $x = - 2$

#### Explanation:

According to the Rational Root Theorem in its simplified form
when $f \left(x\right)$is a monic polynomial (a polynomial whose highest degree term has a coefficient of $1$)
if $a$ is a rational root of $f \left(x\right)$ then $a$ must be a factor of the constant term of $f \left(x\right)$

For the given example, this means that any rational root of ${x}^{4} - 7 {x}^{2} + 12$ but be a factor of $12$.

The factors of $12$ are $\left\{1 , 2 , 3 , 4 , 6 , - 1 , - 2 , - 3 , - 4 , - 6\right\}$

We can test each of these factors (as I did with Excel, below) to find all rational roots of $f \left(x\right)$