# Objective The student will be able to: - PowerPoint PPT Presentation

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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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Objective The student will be able to:

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### ObjectiveThe student will be able to:

factor using difference of squares.

### Factoring ChartThis chart will help you to determine which method of factoring to use.TypeNumber of Terms

1. GCF 2 or more

2. Difference of Squares 2

### Determine the pattern

= 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

### Review: Multiply (x – 2)(x + 2)

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!!

### 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!!!

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?

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?

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?

Yes

Yes

-

Yes

(9a )(9a )

7b

7b

+

### Factor x2 – y2

• (x + y)(x + y)

• (x – y)(x + y)

• (x + y)(x – y)

• (x – y)(x – y)

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?

Yes

Yes

Yes

-

3(5x )(5x )

2

2

+

### Factor 18c2 + 8d2

• 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

• 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!

### ObjectiveThe student will be able to:

factor perfect square trinomials.

### Factoring ChartThis chart will help you to determine which method of factoring to use.TypeNumber of Terms

1. GCF 2 or more

2. Diff. Of Squares 2

3. Trinomials 3

### 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

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

### 1) Factor x2 + 6x + 9

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?

• Is the first term a perfect square?

Yes, a = x

2)Is the last term a perfect square?

Yes, b = 3

• Is the middle term twice the product of the a and b?

Yes, 2ab = 2(x)(3) = 6x

(x + 3)2

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

### 2) Factor y2 – 16y + 64

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?

• Is the first term a perfect square?

Yes, a = y

2)Is the last term a perfect square?

Yes, b = 8

• Is the middle term twice the product of the a and b?

Yes, 2ab = 2(y)(8) = 16y

(y – 8)2

### Factor m2 – 12m + 36

• (m – 6)(m + 6)

• (m – 6)2

• (m + 6)2

• (m – 18)2

### 3) Factor 4p2 + 4p + 1

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?

• Is the first term a perfect square?

Yes, a = 2p

2)Is the last term a perfect square?

Yes, b = 1

• Is the middle term twice the product of the a and b?

Yes, 2ab = 2(2p)(1) = 4p

(2p + 1)2

Perfect Square Trinomials

(a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2

## 4) Factor 25x2 – 110xy + 121y2

(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

### Factor 9k2 + 12k + 4

• (3k + 2)2

• (3k – 2)2

• (3k + 2)(3k – 2)

• I’ve got no clue…I’m lost!

### Factor 2r2 + 12r + 18

• 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!