Factoring - Difference of Squares
What numbers are Perfect
Squares?
Squares
Perfect Squares
2
1 =1
2
2 =4
2
3 =9
2
4 = 16
2
5 = 25
2
6 = 36
1
4
9
16
25
36
49
64
81
100
x
2
x
4
x
9
16
x
x
25
2
2
a −b =
(a − b)(a + b)
Conditions for
Difference of Squares
2
x − 36
Must
be a binomial with
subtraction.
First term must be a perfect
square. (x)(x) = x2
Second term must be a perfect
square (6)(6) = 36
(x − 6)(x + 6)
Recognizing the Difference of
Squares
2
9m − 49 = (3m + 7)(3m − 7)
Must be a binomial with subtraction.
First term must be a perfect square
(3m)(3m) = 9m2
Second term must be a perfect square
(7)(7) = 49
Check for GCF.
Sometimes it is necessary to remove the GCF
before it can be factored more completely.
2
5 x − 45 y
(
2
5 x −9y
2
2
)
5( x − 3 y )( x + 3 y )
You Try
Difference of Squares
4 x − 25 = (2 x + 5)(2 x − 5)
2
2 x − 8 = 2(x − 4) = 2(x − 2)(x + 2)
2
b + 100 Not the difference of perfect squares
2
2
y − 16 = ( y + 4)( y − 4)
2
Removing a GCF of -1.
In some cases removing a GCF of negative one
will result in the difference of squares.
2
− x + 16
(
2
− 1 x − 16
)
− 1( x − 4 )( x + 4 )
Multiply the following two polynomials: (x + 3)(x+3).
x+3
x+3
x2
x
x
x
x
1
1
1
x
1
1
1
x
1
1
1
( x + 3)
2
A perfect square trinomial having
the form
(a + b)2 = a2 + 2ab + b2
= x 2 + 2( x )( 3) + 9
= x 2 + 6x + 9
Multiply the following two polynomials: (x - 4)(x - 4).
x-4
x2
x-4
-x -x -x -x
-x
1
1
1
1
-x
1
1
1
1
-x
1
1
1
1
-x
1
1
1
1
( x − 4)
2
A perfect square trinomial having
the form
(a - b)2 = a2 - 2ab + b2
2
= x − 2( x)( 4) + ( −4)
2
Step 1:
Factor out the greatest common factor (GCF) if it is
larger than 1.
Step 2:
Determine if two of the terms are perfect squares.
Step 3:
Determine if the remaining term is equal to twice the
factors of the other two terms.
Example:
Factor 4x2 – 20x + 25.
4x2 is a perfect square. 4x2 = (2x)2.
25 is a perfect square.
25 = (5)2.
2(2x)(5) = 20x. 20x is the middle term.
Therefore, 4x2 – 20x + 25 = (2x – 5)2.
c2 – 12c + 36
c2 is a perfect square. c2 = c x c
36 is a perfect square. 36 = -6 x -6
2(c)(-6) = -12c.
This is a perfect square trinomial.
c2 – 12c + 36 = (c-6)2
25a2 – 90ac + 81c2
25a2 is a perfect square. 25a2 = 5a x 5a
81c2 is a perfect square. 81c2 = (-9c) x (-9c)
2(5a)(-9c) = -90ac
This is a perfect square trinomial.
25a2 – 90ac + 81c2 = (5a – 9c)2
3a2 + 24a + 48
The GCF for each term is 3. So first, factor out the 3.
2
2
3a + 24 a + 48 = 3( a + 8 a + 16 )
The first and last terms are perfect squares.
a • a = a2
4 • 4 = 16
2(a)(4) = 8a
Since the first and last terms are perfect squares and the
middle term is equal to twice the product of the factor of
the perfect squares, this is a perfect square trinomial.
2
3a + 24 a + 48 = 3( a + 4 )
2