Rewriting Polynomials by Factoring

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 ${\color{#FF0000}{x}}.$ The result of factoring out the greatest common factor from the given expression is a quadratic expression with a leading coefficient of $1.$ $\begin{gathered} x\left({\color{#009600}{x^2-2x-24}}\right) \end{gathered}$ 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 $c.$ $\begin{aligned} \textbf{General Expression:}&\ x^2+\textcolor{#ff00ff}{b}x+\textcolor{#ff8c00}{c} \\ \textbf{Our Expression:}&\ x^2\textcolor{#ff00ff}{-2}x \textcolor{#ff8c00}{-24} \end{aligned}$ Next, we have to find a factor pair of $\textcolor{#ff8c00}{c}$ $=$ $\textcolor{#ff8c00}{\text{-} 24}$ whose sum is $\textcolor{#ff00ff}{b}$ $=$ $\textcolor{#ff00ff}{\text{-} 2}.$ 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 $\text{-}24$	$\textcolor{#ff8c00}{\text{-}24}$	$\text{-} 23$
$\text{-}1$ and $24$	$\textcolor{#ff8c00}{\text{-}24}$	$23$
$2$ and $\text{-}12$	$\textcolor{#ff8c00}{\text{-}24}$	$\text{-} 10$
$\text{-}2$ and $12$	$\textcolor{#ff8c00}{\text{-}24}$	$10$
$3$ and $\text{-}8$	$\textcolor{#ff8c00}{\text{-}24}$	$\text{-} 5$
$\text{-}3$ and $8$	$\textcolor{#ff8c00}{\text{-}24}$	$5$
$4$ and $\text{-}6$	$\textcolor{#ff8c00}{\text{-}24}$	$\textcolor{#ff00ff}{\text{-}2}$
$\text{-}4$ and $6$	$\textcolor{#ff8c00}{\text{-}24}$	$2$

The factors whose product is $\textcolor{#ff8c00}{\text{-}24}$ and whose sum is $\textcolor{#ff00ff}{\text{-}2}$ are ${\color{#0000FF}{4}}$ and $\textcolor{deepskyblue}{\text{-} 6}.$ With this information, we can now factor the trinomial. $\begin{gathered} x^2\textcolor{#ff00ff}{-2}x\textcolor{#ff8c00}{-24}\quad\Leftrightarrow\quad (x+{\color{#0000FF}{4}})(x\textcolor{deepskyblue}{-6}) \end{gathered}$ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $\begin{gathered} x^3-2x^2-24x\quad \Leftrightarrow \quad x(x+4)(x-6) \end{gathered}$