Factoring perfect square trinomials

Before we explain the straightforward way of factoring perfect square trinomials, we need to define the expression perfect square trinomial

Whenever you multiply a binomial by itself twice, the resulting trinomial is called a perfect square trinomial

For example, (x + 1) × (x + 1) = x² + x + x + 1 = x² + 2x + 1 and x² + 2x + 1 is a perfect square trinomial

Another example is (x − 5) × (x − 5)

(x − 5) × (x − 5) = x² + -5x + -5x + 25 = x² + -10x + 25 and x² + -10x + 25 is a perfect square trinomial

Now, we are ready to start factoring perfect square trinomials

The model to remember when factoring perfect square trinomials is the following:

a² + 2ab + b² = (a + b)² and (a + b)² is the factorization form for a² + 2ab + b²

Notice that all you have to do is to use the base of the first term and the last term

In the model just described,

the first term is a² and the base is a

the last term is b² and the base is b

Put the bases inside parentheses with a plus between them    (a + b)

Raise everything to the second power   (a + b)² and you are done

Notice that I put a plus between a and b. You will put a minus if the second term is negative!

a² + -2ab + b² = (a − b)²

Remember that a² − 2ab + b² = a² + -2ab + b² because a minus is the same thing as adding the negative ( − = + -) So, a² − 2ab + b² is also equal to (a − b)²

Example #1:

Factor x² + 2x + 1

Notice that x² + 2x + 1 = x² + 2x + 1²

Using x² + 2x + 1², we see that... the first term is x² and the base is x

the last term is 1² and the base is 1

Put the bases inside parentheses with a plus between them    (x + 1)

Raise everything to the second power   (x + 1)² and you are done

Example #2:

Factor x² + 24x + 144

But wait before we continue, we need to establish something important when factoring perfect square trinomials.

. How do we know when a trinomial is a perfect square trinomial?

This is important to check this because if it is not, we cannot use the model described above

Think of checking this as part of the process when factoring perfect square trinomials

We will use example #2 to show you how to check this

Start the same way you started example #1:

Notice that x² + 24x + 144 = x² + 24x + 12²

Using x² + 24x + 12², we see that...

the first term is x² and the base is x

the last term is 12² and the base is 12

Now, this is how you check if x² + 24x + 12² is a perfect square

If 2 times (base of first term) times (base of last term) = second term, the trinomial is a perfect square

If the second term is negative, check using the following instead

-2 times (base of first term) times (base of last term) = second term

Since the second term is 24x and 2 × x × 12 = 24x, x² + 24x + 12² is perfect and we factor like this

Put the bases inside parentheses with a plus between them    (x + 12)

Raise everything to the second power   (x + 12)² and you are done

Example #3:

Factor p² + -18p + 81

Notice that p² + -18p + 81 = p² + -18p + 9²

Using p² + -18p + 9², we see that...

the first term is p² and the base is p

the last term is 9² and the base is 9

Since the second term is -18p and -2 × p × 9 = -18p, p² + -18p + 9² is a perfect square and we factor like this

Put the bases inside parentheses with a minus between them    (p − 9)

Raise everything to the second power   (p − 9)² and you are done

Example #4:

Factor 4y² + 48y + 144

Notice that 4y² + 48y + 144 = (2y)² + 48y + 12²

(2y)² + 48y + 12², we see that...

the first term is (2y)² and the base is 2y

the last term is 12² and the base is 12

Since the second term is 48y and 2 × 2y × 12 = 48y, (2y)² + 48p + 12² is a perfect square and we factor like this

Put the bases inside parentheses with a plus between them    (2y + 12)

Raise everything to the second power   (2y + 12)² and you are done

I hope the process illustrated above when factoring perfect square trinomials was easy to follow. Any questions? Send me an email here.