ObjectiveThe student will be able to:

• factor using difference of squares

• factor perfect square trinomials

Difference of SquaresFactoring Questions to ask

Type Number of Terms

1. GCF 2 or more

2. Difference of Squares 2

Determine the pattern1

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 squares1, 4, 9, 16, 25, 36, 49, 64, 81,

100, 121, 144, 169, 196, 225…

Difference of Squares

a2 - b2 = (a - b)(a + b)or

a2 - b2 = (a + b)(a - b)

The order does not matter!!

4 Steps for factoringDifference 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 - 25Do 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!

No

Yes

x2 – 25

Yes

Yes

Yes

( )( )5 xx + 5-

2. Factor 16x2 - 9

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!

No

Yes16x2 – 9

Yes

Yes

Yes

(4x )(4x )3+ 3-

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!

(9a )(9a )7b+ 7b-

3. Factor 81a2 – 49b2

No

Yes 81a2 – 49b2

Yes

Yes

Yes

Factor x2 – y2

(x – y)(x + y)

(x + y)(x – y)OR

Multiplication is communitive order doesn’t matter

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! 3(5x )(5x )2+ 2-

4. Factor 75x2 – 12

Yes! GCF = 3

Yes 3(25x2 – 4)

Yes

Yes

Yes

Factor 18c2 + 8d2

You cannot factor using difference of squares because there is no subtraction!

GCF = 22(9c2 + 4d2)

Two terms?1st term a perfect square?2nd term a perfect square?Subtraction?

YESYESYESNO!!!

Final Answer: 2(9c2 + 4d2)

Factor -64 + 4m2

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

Factor => 4(m -4 )(m + 4)

Perfect Square TrinomialsFactoring Questions

Type Number of Terms

1. GCF 2 or more

2. Diff. Of Squares 2

3. Trinomials 3

Perfect Square Trinomials(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

y2 + 4y + 4

y2

+2y+2y+4

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

Using the formula, (y + 2)2 = (y)2 + 2(y)(2) + (2)2

(y + 2)2 = y2 + 4y + 4

Which one is quicker?

1) Factor x2 + 6x + 9

Does this fit the form of our perfect square trinomial?

1) Is the first term a perfect square?

Yes, a = x2) Is the last term a perfect

square?Yes, b = 3

3) Is the middle term twice the product of the a and b?Yes, 2ab = 2(x)(3) = 6x

Perfect Square Trinomials(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your answer!

(x + 3)2

You can still factor the other way but this is quicker!

2) Factor y2 – 16y + 64

Does this fit the form of our perfect square trinomial?

1) Is the first term a perfect square?

Yes, a = y2) Is the last term a perfect

square?Yes, b = 8

3) Is the middle term twice the product of the a and b?Yes, 2ab = 2(y)(8) = 16y

Perfect Square Trinomials(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your answer!

(y – 8)2

Factor m2 – 12m + 36

(m - 6)(m - 6 )

Any GCF? NO!!

a and c perfect squares? YES!!

Factors of first term = m * m Factors of last term = 6 * 6

3) Factor 4p2 + 4p + 1

Does this fit the form of our perfect square trinomial?

1) Is the first term a perfect square?

Yes, a = 2p2) Is the last term a perfect

square?Yes, b = 1

3) Is the middle term twice the product of the a and b?Yes, 2ab = 2(2p)(1) = 4p

Perfect Square Trinomials(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your answer!

(2p + 1)2

Does this fit the form of our perfect square trinomial?

1) Is the first term a perfect square?

Yes, a = 5x2) Is the last term a perfect

square?Yes, b = 11y

3) Is the middle term twice the product of the a and b?

Yes, 2ab = 2(5x)(11y) = 110xy

4) Factor 25x2 – 110xy + 121y2

Perfect Square Trinomials(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your answer!

(5x – 11y)2

Factor 9k2 + 12k + 4

1. (3k + 2)2

2. (3k – 2)2

3. (3k + 2)(3k – 2)

4. I’ve got no clue…I’m lost!

Factor 2r2 + 12r + 18

1. prime

2. 2(r2 + 6r + 9)

3. 2(r – 3)2

4. 2(r + 3)2

5. 2(r – 3)(r + 3)

Don’t forget to factor the GCF first!

Conditions for Difference of Squares

• 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

362 x

66 xx

Recognizing the Difference of Squares

)10)(10( pp1002 pMust 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

)73)(73( mm499 2 mMust 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.

22 455 yx 22 95 yx

yxyx 335

Removing a GCF of -1.

In some cases removing a GCF of negative one will result in the difference of squares.

162 x

161 2 x

441 xx