Now, we are ready to start factoring perfect square trinomials
The model to remember when factoring perfect square trinomials is the following:
a^{2} + 2ab + b^{2} = (a + b)^{2} and (a + b)^{2} is the factorization form for a^{2} + 2ab + b^{2}
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 a^{2} and the base is a
the last term is b^{2} 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!
a^{2} + -2ab + b^{2} = (a − b)^{2}
Remember that a^{2} − 2ab + b^{2} = a^{2} + -2ab + b^{2} because a minus is the
same thing as adding the negative ( − = + -)
So, a^{2} − 2ab + b^{2} is also equal to (a − b)^{2}
Example #1:
Factor x^{2} + 2x + 1
Notice that x^{2} + 2x + 1 = x^{2} + 2x + 1^{2}
Using x^{2} + 2x + 1^{2}, we see that...
the first term is x^{2} and the base is x
the last term is 1^{2} 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 x^{2} + 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