Transcript Factoring Special Cases
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