Download Factor Difference of Squares

Objective
The student will be able to:
• factor using difference of squares
• factor perfect square trinomials
Difference of Squares
Factoring Questions to ask
Type
Number of Terms
1. GCF
2. Difference of Squares
2 or more
2
Determine the pattern
1
4
9
16
25
36
…
= 12
= 22
= 32
= 42
= 52
= 62
These are perfect squares!
You should be able to list
the first 15 perfect
squares in 30 seconds…
Perfect squares
1, 4, 9, 16, 25, 36, 49, 64, 81,
100, 121, 144, 169, 196, 225…
Difference of Squares
2
2
a - b = (a - b)(a + b)
or
2
2
a - b = (a + b)(a - b)
The order does not matter!!
4 Steps for factoring
Difference of Squares
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!!!
1. Factor x2 - 25
Do you have a GCF? No
Are the Difference of Squares steps true?
Two terms? Yes
2 – 25
st
Yes
x
1 term a perfect square?
2nd term a perfect square? Yes
Subtraction? Yes
Write your answer!
( x + 5 )(x - 5 )
2. Factor 16x2 - 9
Do you have a GCF? No
Are the Difference of Squares steps true?
Two terms? Yes
2–9
16x
1st term a perfect square? Yes
2nd term a perfect square?Yes
Subtraction? Yes
(4x
+
3
)(4x
3)
Write your answer!
3. Factor 81a2 – 49b2
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
81a2 – 49b2
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
(9a + 7b)(9a - 7b)
Write your answer!
Factor
2
x
–
2
y
(x – y)(x + y)
OR
(x + y)(x – y)
Multiplication is communitive
 order doesn’t matter
4. Factor
2
75x
– 12
When factoring, use your factoring table.
Do you have a GCF? Yes! GCF = 3
3(25x2 – 4)
Are the Difference of Squares steps true?
Two terms? Yes
3(25x2 – 4)
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
3(5x + 2 )(5x - 2 )
Write your answer!
Factor 18c2 + 8d2
GCF = 2
2(9c2 + 4d2)
Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
YES
YES
YES
NO!!!
You cannot factor using difference of squares
because there is no subtraction!
Final Answer: 2(9c2 + 4d2)
Factor -64 +
2
4m
Rewrite the problem as 4m2 – 64 so the
subtraction is in the middle!
4m2 – 64
GCF? Yes => 4
Is the leftover the difference
of perfect squares?? YES
4(m -4 )(m + 4)
Factor =>
Perfect Square Trinomials
Factoring Questions
Type
Number of Terms
1. GCF
2 or more
2. Diff. Of Squares 2
3. Trinomials
3
Perfect Square Trinomials
2
2
2
(a + b) = a + 2ab + b
(a - b)2 = a2 – 2ab + b2
Review: Multiply (y + 2)2
(y + 2)(y + 2)
Do you remember these?
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
y2
Using the formula,
+2y (y + 2)2 = (y)2 + 2(y)(2) + (2)2
2 = y2 + 4y + 4
(y
+
2)
+2y
+4
Which one is quicker?
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
y2 + 4y + 4
1) Factor x2 + 6x + 9
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = x
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(x + 3)2
Yes, b = 3
3) Is the middle term twice the
You can still
product of the a and b?
factor the other way
but this is quicker!
Yes, 2ab = 2(x)(3) = 6x
2) Factor y2 – 16y + 64
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = y
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(y – 8)2
Yes, b = 8
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(y)(8) = 16y
Factor m2 – 12m + 36
Any GCF? NO!!
a and c perfect squares? YES!!
Factors of first term = m * m
Factors of last term = 6 * 6
(m - 6)(m - 6 )
3) Factor 4p2 + 4p + 1
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 2p
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(2p + 1)2
Yes, b = 1
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(2p)(1) = 4p
4) Factor 25x2 – 110xy + 121y2
Does this fit the form of our
Perfect Square Trinomials
perfect square trinomial?
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 5x
Since all three are true,
2) Is the last term a perfect
write your answer!
square?
(5x – 11y)2
Yes, b = 11y
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(5x)(11y) = 110xy
Factor 9k2 + 12k + 4
1.
2.
3.
4.
(3k + 2)2
(3k – 2)2
(3k + 2)(3k – 2)
I’ve got no
clue…I’m lost!
Factor 2r2 + 12r + 18
1.
2.
3.
4.
5.
prime
2(r2 + 6r + 9)
2(r – 3)2
2(r + 3)2
2(r – 3)(r + 3)
Don’t forget to factor the
GCF first!
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
Recognizing the Difference of
Squares
p  100  ( p  10)( p  10)
2
Must be a binomial with subtraction.
First term must be a perfect square
(p)(p) = p2
Second term must be a perfect square
(10)(10) = 100
Recognizing the Difference of
Squares
9m  49  (3m  7)(3m  7)
2
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.
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