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).

Factor the trinomial as a product of two binomials.

**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:

Factor the trinomial as a product of two binomials.

**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:

Factor the trinomial as a product of two binomials.

**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:

Factor the trinomial as a product of two binomials.

**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:**

**Step 5:**

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