Factoring Polynomials

Let's start by factoring out the greatest common factor. Then, we will factor the resulting trinomial.

Factor Out the Greatest Common Factor

The greatest common factor is a common factor of all the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the greatest common factor is $x.$ The result of factoring out the greatest common factor from the given expression is a quadratic expression with a leading coefficient of $1.$ $$\begin{array}{c}x\left(x^2-2x-24\right)\end{array}$$ Let's temporarily only focus on the expression in parentheses, and bring back the greatest common factor after factoring.

Factor the Quadratic Expression

To factor a quadratic expression with a leading coefficient of $1,$ we first need to identify the values of $b$ and $c.$ $$\begin{array}{rl}\text{General Expression:}& x^2+bx+c\\ \text{Our Expression:}& x^2-2x-24\end{array}$$ Next, we have to find a factor pair of $c$ $=$ $\text{-}24$ whose sum is $b$ $=$ $\text{-}2.$ Note that $\text{-}24$ is a negative number, so for the product of the factors to be negative, they must have opposite signs — one positive and one negative.

Factor Pair	Product of Factors	Sum of Factors
$1$ and $\text{-}24$	$\text{-}24$	$\text{-}23$
$\text{-}1$ and $24$	$\text{-}24$	$23$
$2$ and $\text{-}12$	$\text{-}24$	$\text{-}10$
$\text{-}2$ and $12$	$\text{-}24$	$10$
$3$ and $\text{-}8$	$\text{-}24$	$\text{-}5$
$\text{-}3$ and $8$	$\text{-}24$	$5$
$4$ and $\text{-}6$	$\text{-}24$	$\text{-}2$
$\text{-}4$ and $6$	$\text{-}24$	$2$

The factors whose product is $\text{-}24$ and whose sum is $\text{-}2$ are $4$ and $\text{-}6.$ With this information, we can now factor the trinomial. $$\begin{array}{c}{x}^2-2x-24\iff (x+4)(x-6)\end{array}$$ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $$\begin{array}{c}{x}^3-2x^2-24x\iff x(x+4)(x-6)\end{array}$$