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Rewriting Polynomials by Factoring

Rewriting Polynomials by Factoring 1.10 - Solution

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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 2m4.{\color{#FF0000}{2m^4}}.
2m624m5+64m42m^6-24m^5+64m^4
2m4m22m412m+2m432{\color{#FF0000}{2m^4}}\cdot m^2-{\color{#FF0000}{2m^4}}\cdot 12m+{\color{#FF0000}{2m^4}}\cdot32
2m4(m212m+32){\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 1.1. 2m4(m212m+32)\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.

Factor the Quadratic Expression

To factor a quadratic expression with a leading coefficient of 1,1, we first need to identify the values of bb and c.c. General Expression: m2+bm+cOur Expression: m212m+32\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 c\textcolor{#ff8c00}{c} == 32\textcolor{#ff8c00}{32} whose sum is b\textcolor{#ff00ff}{b} == -12.\textcolor{#ff00ff}{\text{-} 12}. Note that 3232 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.

Factor Pair Product of Factors Sum of Factors
11 and 3232 32\textcolor{#ff8c00}{32} 3333
-1\text{-}1 and -32\text{-}32 32\textcolor{#ff8c00}{32} -33\text{-} 33
22 and 1616 32\textcolor{#ff8c00}{32} 1818
-2\text{-}2 and -16\text{-}16 32\textcolor{#ff8c00}{32} -18\text{-} 18
44 and 88 32\textcolor{#ff8c00}{32} 1212
-4\text{-}4 and -8\text{-}8 32\textcolor{#ff8c00}{32} -12\textcolor{#ff00ff}{\text{-}12}

The factors whose product is 32\textcolor{#ff8c00}{32} and whose sum is -12\textcolor{#ff00ff}{\text{-}12} are -4{\color{#0000FF}{\text{-} 4}} and -8.\textcolor{deepskyblue}{\text{-} 8}. With this information, we can now factor the trinomial. m212m+32(m4)(m8)\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. 2m624m5+64m42m4(m4)(m8)\begin{gathered} 2m^6-24m^5+64m^4 \quad \Leftrightarrow \quad 2m^4(m-4)(m-8) \end{gathered}