You are undoubtedly familiar with the expression and have already known it. In a previous lesson, we saw that the expression is called a quadratic trinomial, and we saw how to factor this expression using a very easy method.

In this chapter, we will also try to work out the expression , but in the case of , which is also an easy case, requiring only a little concentration. All you have to do is follow the basic steps which we will see in detail with many illustrative examples. In order to solidify this process, I suggest you re-do the examples again.

The method used in this lesson to factor the trinomial is called the ac method.

These steps are very simplified, you can skip some of them or adopt your own method when you master the skill of factoring.

Step 1:

At this stage, we multiply the basic factor by the last factor , and we get a number we call . Then,

Then we decompose the number into the product of two numbers: and so that

Step 2:

At this step, after we have defined the two numbers and in the first stage, we substitute by into the expression , and we get:

Step 3:

At this step, we will group the numbers and in two different brackets, between them the addition (+) operation (in some cases we will have to put a minus operation instead of the addition operation)

Step 4:

We factor the two expressions in parentheses and we get a common factor. If you do not get a common factor, then you have probably missed one of the above steps and must recalculate.

Step 5:

And finally, after we defined the common factor in the previous step, we now factor out the trinomial using this common factor. (This common factor must be a binomial expression in parentheses).

## Examples of How to Factor a Trinomial

#### Example1:

Factor the trinomial as a product of two binomials.

#### Solution:

Step 1:

We calculate the product . we have

After we get the number , we re-decompose it into the product of two numbers where their sum is 10.

Step 2:

We substitute by :

Step 3:

Grouping the two numbers and in different parentheses.

Step 4:

The prenthese has the greatest common factor of , and the prenthese has the greatest common factor of

Step 5:

We found that the common factor is , perform one last factorization to get to the final answer:

#### Example2:

Factor the trinomial as a product of two binomials.

#### Solution:

Step 1:

We calculate the product . we have

After we get the number , we re-decompose it into the product of two numbers where their sum is 14.

Step 2:

We substitute by :

Step 3:

Grouping the two numbers and in different parentheses.

Step 4:

The prenthese has the greatest common factor of , and the prenthese has the greatest common factor of

Step 5:

We found that the common factor is , perform one last factorization to get to the final answer:

#### Example3:

Factor the trinomial as a product of two binomials.

#### Solution:

Step 1:

We calculate the product . we have

After we get the number , we re-decompose it into the product of two numbers where their sum is -13.

Step 2:

We substitute by :

Step 3:

Grouping the two numbers and in different parentheses.

Step 4:

The prenthese has the greatest common factor of , and the prenthese has the greatest common factor of

Step 5:

We found that the common factor is , perform one last factorization to get to the final answer:

#### Example5:

Factor the trinomial as a product of two binomials.

#### Solution:

Step 1:

We calculate the product . we have

After we get the number , we re-decompose it into the product of two numbers where their sum is 11.

Step 2:

We substitute by :

Step 3:

Grouping the two numbers and in different parentheses.

Step 4:

The prenthese has the greatest common factor of , and the prenthese has the greatest common factor of

Step 5:

We found that the common factor is , perform one last factorization to get to the final answer: