# Factoring

## Factor Trinomials


### Factor Trinomials of the Form 

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication. To factor the trinomial means to start with the product, and end with the factors.
To figure out how we would factor a trinomial of the form  such as  and factor it to  let’s start with two general binomials of the form  and 


This tells us that to factor a trinomial of the form  we need two factors  and  where the two numbers m and n multiply to c and add to b.
Let’s summarize the steps we used to find the factors.
In the first example, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.
How do you get a positive product and a negative sum? We use two negative numbers.
Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.
How do you get a negative product and a positive sum? We use one positive and one negative number.
When we factor trinomials, we must have the terms written in descending order—in order from highest degree to lowest degree.
Sometimes you’ll need to factor trinomials of the form  with two variables, such as  The first term,  is the product of the first terms of the binomial factors,  The  in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in How To Factor trinomials.
Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.
Let’s summarize the method we just developed to factor trinomials of the form 

### Factor Trinomials of the form ax2 + bx + c using Trial and Error

Our next step is to factor trinomials whose leading coefficient is not 1, trinomials of the form 
Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods we’ve used so far. Let’s do an example to see how this works.
What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.
Let’s factor the trinomial 
From our earlier work, we expect this will factor into two binomials.
We know the first terms of the binomial factors will multiply to give us  The only factors of  are  We can place them in the binomials.
Check: Does 
We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1, 2. But we now have two cases to consider as it will make a difference if we write 1, 2 or 2, 1.
Which factors are correct? To decide that, we multiply the inner and outer terms.
Since the middle term of the trinomial is  the factors in the first case will work. Let’s use FOIL to check.
Our result of the factoring is:
Remember, when the middle term is negative and the last term is positive, the signs in the binomials must both be negative.
When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.
Don’t forget to look for a GCF first and remember if the leading coefficient is negative, so is the GCF.

### Factor Trinomials of the Form  using the “ac” Method

Another way to factor trinomials of the form  is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!
The “ac” method is summarized here.
Don’t forget to look for a common factor!

### Factor 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 u for the substitution.
In the  the middle term has a variable, x, and its square,  is the variable part of the first term. Look for this relationship as you try to find a substitution.
Sometimes the expression to be substituted is not a monomial.

### Key Concepts

1. How to factor trinomials of the form 


2. Strategy for Factoring Trinomials of the Form  When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.
3. How to factor trinomials of the form 


4. How to factor trinomials of the form 



### Practice Makes Perfect

Factor Trinomials of the Form 
In the following exercises, factor each trinomial of the form 
In the following exercises, factor each trinomial of the form 
Factor Trinomials of the Form 
In the following exercises, factor completely using trial and error.
Factor Trinomials of the Form 
In the following exercises, factor using the ‘ac’ method.
Factor Using Substitution
In the following exercises, factor using substitution.
Mixed Practice
In the following exercises, factor each expression using any method.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?