Bell Ringer 2/20/15

Completely Factor & Check your answer.

1.Factor: 2x2 - 14x + 12

2.Factor: y2 + 4y + 4

3.Factor: 75x2 – 12

ObjectiveThe student will be able to:

factor perfect square trinomials.

Factoring ChartThis chart will help you to determine

which method of factoring to use.Type Number of Terms

1. GCF 2 or more

2. Grouping 4

3. Trinomials 3

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)Check this out…whaaat!!

(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

1. (m – 6)(m + 6)

2. (m – 6)2

3. (m + 6)2

4. (m – 18)2

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

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. Type Number of Terms1. GCF 2 or more

2. Grouping 4

3. Trinomials 3

4. 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 …

Perfect squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Review: Multiply (x – 2)(x + 2)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

x2 – 4

x -2

x

+2

x2

+2x

-2x

-4

This is called the difference of squares.

x2

+2x-2x-4

Notice the middle terms

eliminate each other!

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 - 25When 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!

No

Yes x2 – 25

Yes

Yes

Yes

( )( )5 xx + 5-

2. Factor 16x2 - 9When 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!

No

Yes 16x2 – 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

1. (x + y)(x + y)

2. (x – y)(x + y)

3. (x + y)(x – y)

4. (x – y)(x – y)

Remember, the order doesn’t matter!

Factor 18c2 + 8d2

1. prime

2. 2(9c2 + 4d2)

3. 2(3c – 2d)(3c + 2d)

4. 2(3c + 2d)(3c + 2d)

You cannot factor using difference of squares because there is no

subtraction!

ObjectiveThe student will be able to:

use the zero product property to solve equations

Zero Product Property

If a • b = 0 then

a=0,

b=0,

or both a and b equal 0.

1. Set the equation equal to 0.2. Factor the equation.3. Set each part equal to 0 and

solve.4. Check your answer on the

calculator if available.

4 steps for solving a quadratic equation

Set = 0Factor

Split/SolveCheck

Using the Zero Product Property, you know that either x + 3 = 0 or x - 5 = 0

Solve each equation.

x = -3 or x = 5

{-3, 5}

1. Solve (x + 3)(x - 5) = 0

2. Solve (2a + 4)(a + 7) = 02a + 4 = 0 or a + 7 = 0

2a = -4 or a = -7

a = -2 or a = -7

{-2, -7}

3. Solve (3t + 5)(t - 3) = 0

3t + 5 = 0 or t - 3 = 0

3t = -5 or t = 3

t = -5/3 or t = 3

{-5/3, 3}

4. Solve x2 - 11x = 0

GCF = x

x(x - 11) = 0

x = 0 or x - 11 = 0

x = 0 or x = 11

{0, 11}

Set = 0Factor

Split/SolveCheck