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) = x
^{2} + x + x + 1 = x
^{2} + 2x + 1 and x
^{2} + 2x + 1 is a perfect
square trinomial
Another example is (x − 5) × (x − 5)
(x − 5) × (x − 5) = x
^{2} + -5x + -5x + 25 = x
^{2} + -10x + 25 and x
^{2} + -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:
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
Start the same way you started example #1:
Notice that x
^{2} + 24x + 144 = x
^{2} + 24x + 12
^{2}
Using x
^{2} + 24x + 12
^{2}, we see that...
the first term is x
^{2} and the base is x
the last term is 12
^{2} and the base is 12
Now, this is how you check if x
^{2} + 24x + 12
^{2} 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, x
^{2} + 24x + 12
^{2} 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 p
^{2} + -18p + 81
Notice that p
^{2} + -18p + 81 = p
^{2} + -18p + 9
^{2}
Using p
^{2} + -18p + 9
^{2}, we see that...
the first term is p
^{2} and the base is p
the last term is 9
^{2} and the base is 9
Since the second term is -18p and -2 × p × 9 = -18p, p
^{2} + -18p + 9
^{2} 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 4y
^{2} + 48y + 144
Notice that 4y
^{2} + 48y + 144 = (2y)
^{2} + 48y + 12
^{2}
(2y)
^{2} + 48y + 12
^{2}, we see that...
the first term is (2y)
^{2} and the base is 2y
the last term is 12
^{2} and the base is 12
Since the second term is 48y and 2 × 2y × 12 = 48y, (2y)
^{2} + 48p + 12
^{2} 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.
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