Objective The student will be able to:. factor using difference of squares. Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms. 1. GCF 2 or more 2. Difference of Squares 2. Determine the pattern. = 1 2
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factor using difference of squares.
1. GCF 2 or more
2. Difference of Squares 2
= 12
= 22
= 32
= 42
= 52
= 62
These are perfect squares!
You should be able to list the first 15 perfect squares in 30 seconds…
1
4
9
16
25
36
…
Perfect squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
x2 – 4
Notice the middle terms eliminate each other!
x2
+2x
2x
x2
2x
4
+2x
4
This is called the difference of squares.
Difference of Squares
a2  b2 = (a  b)(a + b)or
a2  b2 = (a + b)(a  b)
The order does not matter!!
1. Are there only 2 terms?
2. Is the first term a perfect square?
3. Is the last term a perfect square?
4. Is there subtraction (difference) in the problem?
If all of these are true, you can factor using this method!!!
No
1. Factor x2  25
x2 – 25
Yes
When factoring, use your factoring table.
Do you have a GCF?
Are the Difference of Squares steps true?
Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!
Yes
Yes

Yes
( )( )
x
+
5
x
5
No
2. Factor 16x2  9
16x2 – 9
Yes
When factoring, use your factoring table.
Do you have a GCF?
Are the Difference of Squares steps true?
Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!
Yes
Yes

Yes
(4x )(4x )
+
3
3
No
3. Factor 81a2 – 49b2
81a2 – 49b2
Yes
When factoring, use your factoring table.
Do you have a GCF?
Are the Difference of Squares steps true?
Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!
Yes
Yes

Yes
(9a )(9a )
7b
7b
+
Remember, the order doesn’t matter!
Yes! GCF = 3
4. Factor 75x2 – 12
Yes
3(25x2 – 4)
When factoring, use your factoring table.
Do you have a GCF?
3(25x2 – 4)
Are the Difference of Squares steps true?
Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!
Yes
Yes
Yes

3(5x )(5x )
2
2
+
You cannot factor using difference of squares because there is no subtraction!
Rewrite the problem as 4m2 – 64 so the subtraction is in the middle!
factor perfect square trinomials.
1. GCF 2 or more
2. Diff. Of Squares 2
3. Trinomials 3
Do you remember these?
(a + b)2 = a2 + 2ab + b2 (a  b)2 = a2 – 2ab + b2
y2
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
y2 + 4y + 4
Using the formula,
(y + 2)2 = (y)2 + 2(y)(2) + (2)2
(y + 2)2 = y2 + 4y + 4
Which one is quicker?
+2y
+2y
+4
Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2 (a  b)2 = a2 – 2ab + b2
Does this fit the form of our perfect square trinomial?
Yes, a = x
2)Is the last term a perfect square?
Yes, b = 3
Yes, 2ab = 2(x)(3) = 6x
Since all three are true, write your answer!
(x + 3)2
You can still factor the other way but this is quicker!
Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2 (a  b)2 = a2 – 2ab + b2
Does this fit the form of our perfect square trinomial?
Yes, a = y
2)Is the last term a perfect square?
Yes, b = 8
Yes, 2ab = 2(y)(8) = 16y
Since all three are true, write your answer!
(y – 8)2
Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2 (a  b)2 = a2 – 2ab + b2
Does this fit the form of our perfect square trinomial?
Yes, a = 2p
2)Is the last term a perfect square?
Yes, b = 1
Yes, 2ab = 2(2p)(1) = 4p
Since all three are true, write your answer!
(2p + 1)2
Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2 (a  b)2 = a2 – 2ab + b2
4) Factor 25x2 – 110xy + 121y2
Since all three are true, write your answer!
(5x – 11y)2
Does this fit the form of our perfect square trinomial?
Is the first term a perfect square?
Yes, a = 5x
Is the last term a perfect square?
Yes, b = 11y
Is the middle term twice the product of the a and b?
Yes, 2ab = 2(5x)(11y) = 110xy
Don’t forget to factor the GCF first!