Let's start by factoring out the . Then, we will the resulting .
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 and the greatest . In this case, the greatest common factor is
x.
x3−2x2−24x x⋅x2−x⋅2x−x⋅24 x(x2−2x−24)
The result of factoring out the greatest common factor from the given expression is a with a of
1.
x(x2−2x−24)
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, we first need to identify the values of b and c.
General Expression:Our Expression: x2+bx+c x2−2x−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. x2−2x−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.
x3−2x2−24x⇔x(x+4)(x−6)