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# Rewriting Polynomials by Factoring

## Rewriting Polynomials by Factoring 1.10 - Solution

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 ${\color{#FF0000}{2m^4}}.$
$2m^6-24m^5+64m^4$
${\color{#FF0000}{2m^4}}\cdot m^2-{\color{#FF0000}{2m^4}}\cdot 12m+{\color{#FF0000}{2m^4}}\cdot32$
${\color{#FF0000}{2m^4}}\left(m^2-12m+32\right)$
The result of factoring out the greatest common factor from the given expression is a quadratic expression with a leading coefficient of $1.$ $\begin{gathered} 2m^4\left({\color{#009600}{m^2-12m+32}}\right) \end{gathered}$ Let's temporarily only focus on the expression in parentheses, and bring back the greatest common factor after factoring.

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:}&\ m^2+\textcolor{#ff00ff}{b}m+\textcolor{#ff8c00}{c} \\ \textbf{Our Expression:}&\ m^2\textcolor{#ff00ff}{-12}m +\textcolor{#ff8c00}{32} \end{aligned} Next, we have to find a factor pair of $\textcolor{#ff8c00}{c}$ $=$ $\textcolor{#ff8c00}{32}$ whose sum is $\textcolor{#ff00ff}{b}$ $=$ $\textcolor{#ff00ff}{\text{-} 12}.$ Note that $32$ is a positive number, so for the product of the factors to be positive, they must have the same sign — both positive or both negative.
$1$ and $32$ $\textcolor{#ff8c00}{32}$ $33$
$\text{-}1$ and $\text{-}32$ $\textcolor{#ff8c00}{32}$ $\text{-} 33$
$2$ and $16$ $\textcolor{#ff8c00}{32}$ $18$
$\text{-}2$ and $\text{-}16$ $\textcolor{#ff8c00}{32}$ $\text{-} 18$
$4$ and $8$ $\textcolor{#ff8c00}{32}$ $12$
$\text{-}4$ and $\text{-}8$ $\textcolor{#ff8c00}{32}$ $\textcolor{#ff00ff}{\text{-}12}$
The factors whose product is $\textcolor{#ff8c00}{32}$ and whose sum is $\textcolor{#ff00ff}{\text{-}12}$ are ${\color{#0000FF}{\text{-} 4}}$ and $\textcolor{deepskyblue}{\text{-} 8}.$ With this information, we can now factor the trinomial. $\begin{gathered} m^2\textcolor{#ff00ff}{-12}m+\textcolor{#ff8c00}{32}\quad\Leftrightarrow\quad (m{\color{#0000FF}{\,-\,4}})(m-\textcolor{deepskyblue}{8}) \end{gathered}$ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $\begin{gathered} 2m^6-24m^5+64m^4 \quad \Leftrightarrow \quad 2m^4(m-4)(m-8) \end{gathered}$