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 with the correct sign. Several examples demonstrate identifying and factoring perfect square trinomials.

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
 