# How do you factor completely t^3+t^2-22t-40?

Mar 2, 2017

${t}^{3} + {t}^{2} - 22 t - 40 = \textcolor{g r e e n}{\left(t + 2\right) \left(t + 4\right) \left(t - 5\right)}$

#### Explanation:

Using the Rational Root Theorem, the possible roots of the given polynomial are contained in the set:
$\textcolor{w h i t e}{\text{XXX}} \left\{\pm 1 , \pm 2 , \pm 4 , \pm 8 , \pm 10 , \pm 20\right\}$

Evaluating the given polynomial for each possible root (I chose to use a spread sheet to do this; see below)
we can determine the roots: $- 2 , - 4 , \mathmr{and} + 5$
$\textcolor{w h i t e}{\text{XXX}}$note that since the polynomial is of degree 3
$\textcolor{w h i t e}{\text{XXXXX}}$there can be a maximum of 3 unique roots.

which implies the factors:
$\textcolor{w h i t e}{\text{XXX}} \left(t + 2\right) \left(t + 4\right) \left(t - 5\right)$