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}−2x−24 $ Next, we have to find a factor pair of $c$ $=$ $-24$ whose sum is $b$ $=$ $-2.$ Note that $-24$ 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 $-24$ | $-24$ | $-23$ |
$-1$ and $24$ | $-24$ | $23$ |
$2$ and $-12$ | $-24$ | $-10$ |
$-2$ and $12$ | $-24$ | $10$ |
$3$ and $-8$ | $-24$ | $-5$ |
$-3$ and $8$ | $-24$ | $5$ |
$4$ and $-6$ | $-24$ | $-2$ |
$-4$ and $6$ | $-24$ | $2$ |
The factors whose product is $-24$ and whose sum is $-2$ are $4$ and $-6.$ With this information, we can now factor the trinomial. $x_{2}−2x−24⇔(x+4)(x−6) $ 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}−2x_{2}−24x⇔x(x+4)(x−6) $