# How do you use the rational root theorem to find the roots of  f(x)=x^5-3x^2-4?

Aug 21, 2015

(as far as I can tell) $f \left(x\right) = {x}^{5} - 3 {x}^{2} - 4$ does not have any rational roots (and therefore the Rational Root Theorem is of no use in determining the roots)

#### Explanation:

According to the Rational Root Theorem:
any rational roots of $\textcolor{red}{1} {x}^{5} - 3 {x}^{2} - \textcolor{b l u e}{4}$
must be of the form:
$\textcolor{w h i t e}{\text{XXXX")("integer factor of "color(blue)(4))/("integer factor of } \textcolor{red}{1}}$

The only possible rational roots are therefore:
$\textcolor{w h i t e}{\text{XXXX}} \pm 1 , \pm 2 , \pm 4$

$f \left(+ 1\right) = - 6$
$f \left(- 1\right) = - 8$
$f \left(+ 2\right) = 16$
$f \left(- 2\right) = - 48$
$f \left(+ 4\right) = 972$
$f \left(- 4\right) = - 1076$
Since none of these give a value for $f \left(x\right)$ which is equal to $0$
all possible rational roots provided by the Rational Root Theorem are extraneous.