Download 9.8 Lesson

Factoring - Difference of Squares and
Perfect Square Trinomial Patterns
What numbers are Perfect
Squares?
Squares
1 1
Perfect Squares
2
2 4
2
3 9
2
4  16
2
5  25
2
6  36
2
1
4
9
16
25
36
49
64
81
100
x
2
x
4
x
6
x
8
x
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?
x2 – 9 = (x + 3)(x – 3)
The sum of squares will not factor a2+b2
Using FOIL we find the product
of two binomials.
( x  5)( x  5)
 x  5 x  5 x  25
2
 x  25
2
Rewrite the polynomial as the
product of a sum and a difference.
x  25  ( x  5)( x  5)
2
Conditions for
Difference of Squares
x  36
2
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
Check for GCF.
Sometimes it is necessary to remove the GCF
before it can be factored more completely.
5 x  45 y
2

5 x 9y
2
2
2

5x  3 y x  3 y 
Removing a GCF of -1.
In some cases removing a GCF of negative one
will result in the difference of squares.
 x  16
2

 1 x  16
2

1x  4x  4
Difference of Squares
4 x  25  2 x  52 x  5
2
4Try
2 x  8  2x You
2x  2x  2
2
2
b  100  1 b  100
2
2
1st. not a perfect square.
No GCF. PRIME!
y  16   y  4 y  4
2
Factoring a perfect square trinomial
in the form:
a  2ab  b  (a  b)
2
2
a  2ab  b  (a  b)
2
2
2
2
Perfect Square Trinomials can be
factored just like other trinomials
(guess and check), but if you
recognize the perfect squares pattern,
follow the formula!
a  2ab  b  (a  b)
2
2
a  2ab  b  (a  b)
2
2
2
2
2
Ex: x  8x 16
x 
2
Does the middle term
fit the pattern, 2ab?
4 
2
a
b
2  x  4  8x
Yes, the factors are (a + b)2 :
x  8x  16  x  4
2
2
2
Ex: 4x 12x  9
2x 
2
Does the middle term
fit the pattern, 2ab?
3
2
a
b
2 2x  3  12x
Yes, the factors are (a - b)2 :
4x  12x  9  2x  3
2
2