Factoring - Difference of Squares

What is a Perfect Square

What numbers are Perfect Squares?

Squares 1 2  1 2 3 2 2  4  9 4 2 5 2 6 2  16   25 36 Perfect Squares 1 4 9 16 25 36 49 64 81 100

x

2

x x x x

4 6 8 10

Factoring: Difference of Squares Count the number of terms. Is it a binomial?

Is the first term a perfect square?

Is the last term a perfect square?

Is it, or could it be, a subtraction of two perfect squares?

x 2 – 9 = (x + 3)(x – 3) The sum of squares will not factor a 2 +b 2

Using

FOIL

we find the product of two binomials.

(

x

 5 )(

x

 5 ) 

x

2  5

x

 5

x



x

2  25  25

Rewrite the polynomial as the product of a sum and a difference.

x

2  25  (

x

 5 )(

x

 5 )

Conditions for Difference of Squares

x

2  36 Must be a binomial with subtraction. First term must be a perfect square.

(x)(x) = x 2 Second term must be a perfect square (6)(6) = 36 

x

 6 

x

 6 

Check for GCF.

Sometimes it is necessary to remove the GCF before it can be factored more completely.

5 5 

x

 5

x



x

2 3 2 

y

  9 45

y

2

y

 2

x

 3

y



Removing a GCF of -1.

In some cases removing a GCF of negative one will result in the difference of squares.

   1 

x

1

x

 2

x

 2  16  4 

x

16   4 

Difference of Squares 2 4

x b

2

x

2 2    25 8 100   2    2

x



x

2  1   

b

2 5  2   

x

2  100 

x

 5   2 

x

 2  1st.

not a perfect square.

y

2  16   No GCF.

PRIME!

y

 4 

y

 4 

Factoring - Difference of Squares