1. Factoring Perfect Square Trinomials

2. Factoring Perfect Square Trinomials
What is a perfect square?
If a number is squared the result is a perfect square.
Example 22=4
4 is a perfect square.
Other examples:
32=9 or 42=16
9 is a perfect square.
16 is a perfect square.

3. Factoring Perfect Square Trinomials
Here is a list of the perfect squares for the numbers 1-30.
12= = =441
22= = =484
32= = =529
42= = =576
52= = =625
62= = =676
72= = =729
82= = =784
92= = =841
102= = =900

4. Factoring Perfect Square Trinomials
When a variable is raised to an even power it is a perfect square.
Example: (x)(x)= x2
x2 is a perfect square.
(x3)(x3)= x6 or (x5)(x5)= x10
x6 and x10 are both perfect squares.

5. Factoring Perfect Square Trinomials
If a number or a variable is a perfect square the square root of the quantity is the number or variable that was squared to get the perfect square.
Example: Square 9.
9x9 = 81
81 is the perfect square.
9 is the square root of 81.
Example: Square x3
(x3) (x3) = x6 or (x3)2 = x6
x6 is the perfect square.
x3 is the square root of x6

6. Factoring Perfect Square Trinomials
Now we are ready to understand the term- perfect square trinomial.
The trinomial that results from squaring a binomial is a perfect square trinomial.
Example: (x+7)2 = x2+14x+49
x2+14x+49 is a perfect square trinomial.
We know that a perfect square trinomial always results when a binomial is squared.
The reverse is also true.
When we factor a perfect square trinomial the result is always a squared binomial.

7. Factoring Perfect Square Trinomials
Here are few examples:
Factor: x2+10x+25
Result: (x+5)2
Check by multiplying
x2+10x+25
Factor: x2+2xy+y2
(x+y) 2
Check by multiplying.
x2+2xy+y2

8. Factoring Perfect Square Trinomials
Not all trinomials are perfect square trinomials.
How do we recognize that a trinomial is a perfect square trinomial.
The first and last terms of the trinomial must be perfect squares and must be positive.
Example: x2+10x+25
What about the middle term? +10x
Take the square root of the first term x2 and get x.
Take the square root of the last term +25 and get 5.
Multiply (5)(x) and double the result. 10x. That is your middle term.
Two times the product of the square roots of the first and last terms will give the middle term.

9. Factoring Perfect Square Trinomials
Here are some examples of trinomials that are perfect square trinomials.
4x2 -20x +25 2x 5
(2x- 5)2
9x2 - 48xy + 64y2 3x 8y
(3x-8y)2
2x3 +20x2y+50xy2
Factor out the GCF
2x(x2+10xy+25y2) x 5y
2x(x+5y)2

10. Factoring Perfect Square Trinomials
Here are some examples that are not perfect square trinomials.
x2+10x-25
The last term is not positive.
x2+2xy+2y2
The 2 in the last term is not a perfect square.
4x2-10xy+25y2
The square root of the first term is 2x.
The square root of the last term 5y.
2(2x)(5y)= 20xy
20xy = 10xy
4x2-16xy+8y2
There is a common factor of four.
4(x2- 4xy + 2y2)
The last term of the trinomial is not a perfect square because the 2 in the last term is not perfect square.
To get more help go to the tutorial Practice- Factoring Perfect Square Trinomials