Activity
Let’s learn a new method that always works on any trinomial of the form .
How to factor trinomials of the form using the “” method:
Step 1 - Factor any GCF. Don’t forget!
Step 2 - Find the product .
Step 3 - Find two numbers, and , that
- multiply to : .
- add to : .
Step 4 - Split the middle term using and .
Step 5 - Factor by grouping.
Step 6 - Check by multiplying the factors.
Factor each trinomial using the “” method.
1.
Compare your answer:
2.
Compare your answer:
3.
Compare your answer:
There will be times when a trinomial does not seem to be in the form .
In some of these cases, we can make a substitution that allows the trinomial to be written in the form. This is called factoring by substitution.
It is standard to use for the substitution. The substitution for might be a variable with an exponent, such as , or it might be a binomial, such as .
Factor each trinomial using the substitution method.
4.
Compare your answer:
5.
Compare your answer:
6.
Compare your answer:
Self Check
Additional Resources
Factoring Trinomials Using the “” Method
Another way to factor trinomials of the form is the “” method. (The “” method is sometimes called the grouping method.) The “” method is actually an extension of the methods you used in the last section to factor trinomials with a leading coefficient of 1. This method is very structured (that is, step by step), and it always works!
Example 1
Factor using the “” method: .
Step 1 - Factor any GCF.
Is there a greatest common factor?
No!
Step 2 - Find the product .
Step 3 - Find two numbers, and , that
- multiply to : .
- add to : .
Find two numbers that multiply to 12 and add to 7. Both factors must be positive.
Step 4 - Split the middle term using and .
Rewrite as . It would give the same result if we used .
Notice that is equal to . We just split the middle term to get a more useful form.
Step 5 - Factor by grouping.
Step 6 - Check by multiplying the factors.
The “” method is summarized here:
How to factor trinomials of the form using the “” method:
Step 1 - Factor any GCF. Don’t forget!
Step 2 - Find the product .
Step 3 - Find two numbers, and , that
- multiply to : .
- add to : .
Step 4 - Split the middle term using and .
Step 5 - Factor by grouping.
Step 6 - Check by multiplying the factors.
Try it
Try It: Factoring Trinomials Using the “” Method
Factor using the “” method: .
Here is how to use the “” method to factor a trinomial.
Step 1 - Is there a greatest common factor?
Yes. The GCF is 5.
Step 2 - Factor it.
Step 3 - The trinomial inside the parentheses has a leading coefficient that is not .
Find the product .
Step 4 - Find two numbers that multiply to and add to .
Step 5 - Split the middle term.
Step 6 - Factor the trinomial by grouping.
Step 7 - Check by multiplying all three factors.
Factoring Trinomials Using Substitution
Sometimes a trinomial does not appear to be in the form. However, we can often make a thoughtful substitution that will allow us to make it fit the form. This is called factoring by substitution. It is standard to use for the substitution.
In the form , the middle term has a variable, , and its square, , is the variable part of the first term. Look for this relationship as you try to find a substitution.
Example 2
Factor by substitution: .
The variable part of the middle term is , and its square, , is the variable part of the first term. (We know . If we let , we can put our trinomial in the form we need to factor it.
Step 1 - Rewrite the trinomial to prepare for the substitution.
Step 2 - Let and substitute.
Step 3 - Factor the trinomial.
Step 4 - Replace with .
Step 5 - Check:
Try it
Try It: Factoring Trinomials Using Substitution
Factor by substitution: .
In this case, the expression to be substituted is not a monomial. Let .
Here is how to factor using substitution.
The binomial in the middle term, , is squared in the first term. If we let and substitute, our trinomial will be in form.
Step 1 - Rewrite the trinomial to prepare for the substitution.
Step 2 - Let and substitute.
Step 3 - Factor the trinomial.
Step 4 - Replace with .
Step 5 - Simplify inside the parentheses.
This could also be factored by first multiplying out the and the , combining like terms, and then factoring. Most students prefer the substitution method.