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: x_{2}+bx+cx_{2}+10x+16 $ Next, we have to find a factor pair of $c$ $=$ $16$ whose sum is $b$ $=$ $10.$ Note that $16$ 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 $16$ | $16$ | $17$ |
$-1$ and $-16$ | $16$ | $-17$ |
$2$ and $8$ | $16$ | $10$ |
$-2$ and $-8$ | $16$ | $-10$ |
$4$ and $4$ | $16$ | $8$ |
$-4$ and $-4$ | $16$ | $-8$ |
The factors whose product is $16$ and whose sum is $10$ are $2$ and $8.$ With this information, we can now factor the trinomial. $x_{2}+10x+16⇔(x+2)(x+8) $ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $x_{3}+10x_{2}+16x⇔x(x+2)(x+8) $