Let's start by factoring out the . Then, we will factor 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+10x2+16x x⋅x2+x⋅10x+x⋅16 x(x2+10x+16)
The result of factoring out the greatest common factor from the given expression is a with a of
1.
x(x2+10x+16)
Let's temporarily only focus on the expression in parentheses, and bring back the greatest common factor after factoring.
Factor the Quadratic Expression
To 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+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. x2+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.
x3+10x2+16x⇔x(x+2)(x+8)