What mathematical pattern can be seen in a perfect square trinomial and how is the
pattern used to factor the 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