In the polynomial lesson, we saw that a trinomial is a special case of a polynomial, and a trinomial means an **algebraic expression of three terms**. In another lesson, we saw how to factor polynomials using the common factor (out of the gcd of the polynomial terms).

In this lesson, we will learn about how to factor a quadratic trinomial.

From the title of the lesson, you noticed that we called it “simple quadratic trinomial”, because we will take in the trinomial expression as a special case and we get . The factoring will be very easy in this case. Just read the lesson to the end and re-do the examples.

In another lesson, we will explain how to factor a quadratic trinomial with .

To factor the quadratic trinomial, we need to find the numbers and . Below we will see how to find them.

First, let's expand the expression by FOIL method :

After expanding and simplifying the expression, we can conclude that the number is equal to , and the number is equal to the number . Thus, we get a simple system:

**Step1:** The equation gives us two important ideas:

The first idea (The sign of the two numbers and )

If the number is positive, then the two numbers and have the same sign.

If the number is negative, then the two numbers and have different signs.

The second idea (decomposition into product of two numbers)

We do all the possible decompositions of into product of two numbers. Having defined the sign previously, we can now only take the appropriate decompositions.

If is positive, we take all of the decompositions that they factors out have the same sign.

If is negative, we take all of the decompositions that they factors out have two different signs.

**Step2:** The equation will enable us to determine the final decomposition from among the decompositions that we obtained in the first step, and from it we will get the two numbers m and n.

Note that if , we factor the quadratic trinomial by -1, then we apply the previous steps (See Example 5).

Let's factoring the quadratic trinomial:

**Step1:** We decompose the number 15:

**Step2:** We add the two factors found in step 1

Hence,

therefore,

Let's factoring the quadratic trinomial:

**Step1:** We decompose the number 20:

**Step2:** We add the two factors found in step 1

Hence,

therefore,

Let's factoring the quadratic trinomial:

**Step1:** We decompose the number -21:

**Step2:** We add the two factors found in step 1

Hence,

therefore,

Let's factoring the quadratic trinomial:

Let

**Step1:** We decompose the number 8:

**Step2:** We add the two factors found in step 1

Hence,

therefore,

Let's factoring the quadratic trinomial:

In this example, we have . We need to factor the trinomial by . By doing so, the signs of the three terms are switched

**Step1:** We decompose the number 6:

**Step2:** We add the two factors found in step 1

Hence,

therefore,

Note that not all quadratic trinomials can be factored to . For example, trinomial does not factor.