The Differences of Two Squares
For a
2
– b2, do the parentheses:
(
)(
)
...put the first squared thing in front:
(a
)(a
)
...put the second squared thing in back:
(a
b)(a
b)
...and alternate the signs in the middles:
(a – b)(a + b)
•
2
Factor x – 16
2
This is x
– 42, so I get:
x2 – 16 = x2 – 42 = (x – 4)(x + 4)
•
2
Factor 4x – 25
2
This is (2x)
– 52, so I get:
4x2 – 25 = (2x)2 – 52 = (2x – 5)(2x + 5)
•
6
8
Factor 9x – y
This is
(3x3)2 – (y4)2, so I get:
9x6 – y8 = (3x3)2 – (y4)2 = (3x3 – y4)(3x3 + y4)
•
4
Factor x – 1 Copyright © Elizabeth l 2000-2011 All Rights Reserved
2 2
This is (x
) – 12, so I get:
x4 – 1 = (x2)2 – 12 = (x2 – 1)(x2 + 1)
2
Note that I'm not done yet, because x – 1 is itself a difference of squares, so I need to apply the
2
formula again to get the fully-factored form. Since x – 1 = (x – 1)(x + 1), then:
x4 – 1 = (x2)2 – 12 = (x2 – 1)(x2 + 1)
= ((x)2 – (1)2)(x2 + 1)
= (x – 1)(x + 1)(x2 + 1)
The answer to this last exercise depended on the fact that 1, to any power at all, is still just 1.
Warning: Never forget that this formula is for the difference of squares; the sum of squares is always
prime (that is, it can't be factored).
Greatest Common Factor (GCF)
The first step for factoring a polynomial is to "take out" the greatest common factor. There
are two steps to this: find the GCF of the numbers and the GCF of the variables.
Greatest = Largest
Common = Shared
Factor = Factored Piece
3x2 + 6x
Step 1: Factor each term completely.
3•x•x + 3•2•x
Step 2: Find all factors that are in common (the same in all terms)
3•x•x + 3•2•x
Step 3: Pull out the GCF and then divide every term by it
Please remember to put parentheses around the
terms, with the GCF on the outside.
Once you get more advanced, you will probably be
able to do the division in your head.
Step 4: Simplify each term (perform the division)
3x(x + 2)
1) 5x3 - 125x
The GCF is 5x. Take that out, and then divide each term
5x(x2 - 25)
= 5x(x + 5)(x – 5)
2) 4x3 + 6x + 2x2
The GCF is 2x. Again, take that out.
2x(2x2 + 3 + x)
3) 6x2y + 9xy2
The GCF is 3xy. Notice each term has at least one x and y.
3xy(2x + 3y)
“Factoring: Greatest Common Factor and
Difference of Squares”
Directions: Factor the polynomials completely.
1. 18x2y - 8y
1. _______________________________
2. 16x4y2 - 36x2y4
2. _______________________________
3. 25x7y3z2 – 9x5y3z2
3. _______________________________
4. 100x4 - 9y4
4. _______________________________
5. 2x7 – 2x3
5. _______________________________
6. 5x7yz – 20xy3z3
6. _______________________________
7. 12x4y5z4 – 27x2y3z2
7. _______________________________
8. 5x12 – 5x4
8. _______________________________
9. 72a9b6c5 – 50a3b4c7
9. _______________________________
10. 12x7y5z2 + 27x5y3z14
10. ______________________________
Example: Factor the GCF out of a