Let's start by factoring out the greatest common factor. Then, we will factor the resulting trinomial.
To factor a quadratic expression with a leading coefficient of $1,$ we first need to identify the values of $b$ and $c.$ $General Expression:Our Expression: m_{2}+bm+cm_{2}−12m+32 $ Next, we have to find a factor pair of $c$ $=$ $32$ whose sum is $b$ $=$ $-12.$ 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.
Factor Pair | Product of Factors | Sum of Factors |
---|---|---|
$1$ and $32$ | $32$ | $33$ |
$-1$ and $-32$ | $32$ | $-33$ |
$2$ and $16$ | $32$ | $18$ |
$-2$ and $-16$ | $32$ | $-18$ |
$4$ and $8$ | $32$ | $12$ |
$-4$ and $-8$ | $32$ | $-12$ |
The factors whose product is $32$ and whose sum is $-12$ are $-4$ and $-8.$ With this information, we can now factor the trinomial. $m_{2}−12m+32⇔(m−4)(m−8) $ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $2m_{6}−24m_{5}+64m_{4}⇔2m_{4}(m−4)(m−8) $