4/8/2013
Determine the pattern
1
4
9
16
25
36
…
9-5 Difference of Squares and
9-6 Perfect Squares
Review: Multiply (x – 2)(x + 2)
First terms: x2
Outer terms: +2x
Inner terms: -2x
Last terms: -4
Combine like terms.
x2 – 4
Notice the
middle terms
eliminate
each other!
x
-2
x2
-2x
+2 +2x
-4
x
= 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
a2 - b2 = (a - b)(a + b)
or
2
2
a - b = (a + b)(a - b)
The order does not matter!!
This is called the difference of squares.
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? Make sure its in the middle!
If all of these are true, you can factor
using this method!!!
1. Factor x2 - 25
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
Write your answer!
x2 – 25
( x + 5 )(x - 5 )
1
4/8/2013
2. Factor 16x2 - 9
3. Factor 81a2 – 49b2
When factoring, use your factoring table.
When factoring, use your factoring table.
Do you have a GCF? No
Do you have a GCF?
Are the Difference of Squares steps true?
Two terms? Yes
16x2 – 9
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
(4x + 3 )(4x - 3 )
Write your answer!
Are the Difference of Squares steps true?
Two terms? Yes
81a2 – 49b2
1st term a perfect square? Yes
2nd term a perfect square?
Yes
Subtraction? Yes
(9a + 7b)(9a - 7b)
Write your answer!
No
4. Factor 75x2 – 12
Factor x2 – y2
1.
2.
3.
4.
(x + y)(x + y)
(x – y)(x + y)
(x + y)(x – y)
(x – y)(x – y)
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
1st term a perfect square?
2nd term a perfect square?
Subtraction? Yes
Write your answer!
Remember, the order doesn’t matter!
3(25x2 – 4)
Yes
Yes
3(5x + 2 )(5x - 2 )
Factor 18c2 + 8d2
1.
2.
3.
4.
prime
2(9c2 + 4d2)
2(3c – 2d)(3c + 2d)
2(3c + 2d)(3c + 2d)
You cannot factor using
difference of squares
because there is no
subtraction!
Factor -64 + 4m2
1.
2.
3.
4.
prime
(2m – 8)(2m + 8)
4(-16 + m2)
4(m – 4)(m + 4)
Rewrite the problem as
4m2 – 64 so the
subtraction is in the
middle!
2
4/8/2013
1) Factor x2 + 6x + 9
Factor 2 x
4
162
2x4 – 162
= 2(x4 – 81)
= 2(x2 – 9)(x2 + 9)
= 2(x – 3)(x + 3)(x2 + 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
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
Factor m2 – 12m + 36
1.
2.
3.
4.
(m – 6)(m + 6)
(m – 6)2
(m + 6)2
(m – 18)2
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
3
4/8/2013
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!
Each of the following is a perfect
square except
1.
2.
3.
4.
x2 + 8x + 16
4x2 - 20x + 25
9x2 + 12x – 4
All are perfect squares
Videos
1. Factoring Perfect Square Trinomials with a = 1
http://www.phschool.com/atschool/academy123/engl
ish/academy123_content/wl-book-demo/ph280s.html
2. Factoring Perfect Square Trinomials with a ≠ 1
http://www.phschool.com/atschool/academy123/engl
ish/academy123_content/wl-book-demo/ph281s.html
Find the value of c that makes cn2 + 72ny + 81y2 a
perfect square trinomial.
1.
2.
3.
4.
1
4
16
None of these
Homework
9-5 Skills Practice
(1 – 23 odd)
9-6 Skills Practice
(1 – 23 odd)
4