The document discusses factoring perfect square trinomials. A perfect square trinomial is a trinomial that is the result of squaring a binomial. To factor a perfect square trinomial, both the first and last terms must be perfect squares and the middle term must be 2 times the product of the binomial terms. If these conditions are met, the trinomial can be factored by reversing the process of squaring the binomial. Several examples demonstrate how to determine if a trinomial is a perfect square trinomial and how to factor it if so.

1. Factoring - Perfect Square Trinomial
• A Perfect Square Trinomial is any trinomial that is the
result of squaring a binomial.
 
2
3
x 
Binomial
Squared
2
6 9
x x
  
Perfect Square
Trinomial

2. 2 2
2
a ab b
 
• Our goal now is to start with a perfect square trinomial
and factor it into a binomial squared. Here are the
patterns.
Perfect Square
Trinomial
Factored
 
2
a b
 
2 2
2
a ab b
   
2
a b
 
Note the pattern for the signs:

3. • Here is how to identify a perfect square trinomial:
1. Both first and last terms are perfect squares
2 2
2
a ab b
  2 2
2
a ab b
 
2. The middle term is given by 2ab
If these two conditions are met, then the
expression is a perfect square trinomial.
Note that there is always a positive sign on
both of these terms.

4. • Example 1
2
8 16
x x
 
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Is the middle term 2 ?
ab
2
8 16
x x
     
2 2
8 4
x x
  
2ab 2( )(4)
x
 8x


5. • Since the trinomial is a perfect square, factor it using
the pattern:
1. First term a:
2. Last term b:
(x
( 4)
x
3. Sign same as
the middle term
( 4)
x 
4. Squared
2
( 4)
x 
 
2
2 2
2
a ab b a b
   
   
2 2
8 4
x x
 

6. • Example 2
2
10 25
x x
 
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Is the middle term 2 ?
ab
2
10 25
x x
     
2 2
10 5
x x
  
2ab   
2 5
x
 10x


7. • Since the trinomial is a perfect square, factor it using
the pattern:
1. First term:
2. Last term
(x
( 5)
x
3. Sign same as
the middle term
( 5)
x 
4. Squared
2
( 5)
x 
 
2
2 2
2
a ab b a b
   
   
2 2
10 5
x x
 

8. • Example 3
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term:
2 2x
 3

2
4 12 9
x x
 
12x

2ab 

9. • Since the trinomial is a perfect square, factor it using
the pattern:
1. First term:
2. Last term
(2x
(2 3)
x
3. Sign same as
the middle term
(2 3)
x 
4. Squared
2
(2 3)
x 
 
2
2 2
2
a ab b a b
   
2
4 12 9
x x
 

10. • Example 4
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term: 2 2x
 3

2
4 7 9
x x
 
12x
 No
This is not a perfect square trinomial. If it can be
factored, another method will have to be used.

11. • Example 5
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2
9 20 12
x x
 
This is not a perfect square trinomial. If it can be
factored, another method will have to be used.
No

12. • Example 6
Factor:
Determine if the trinomial is a perfect square
trinomial.
2
10 25
x x
 
This is not a perfect square trinomial since the last
term has a negative sign.
Perfect square trinomials always have a positive
sign for the last term.

13. • Example 7
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term:
2 5x
 6y

2 2
25 60 36
x xy y
 
60xy


14. • Since the trinomial is a perfect square, factor it using
the pattern:
1. First term:
2. Last term
(5x
(5 6 )
x y
3. Sign same as
the middle term
(5 6 )
x y

4. Squared
2
(5 6 )
x y

 
2
2 2
2
a ab b a b
   
2 2
25 60 36
x xy y
 