Lesson 10.3 Factoring Trinomials

NCSCOS 1.01; 1.02

Daily ObjectiveTLW factor polynomials by grouping.

TLW will factor trinomials using the “Up and Under” method

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

1. Factor 12ac + 21ad + 8bc + 14bd

Do you have a GCF for all 4 terms? No

Group the first 2 terms and the last 2 terms.

(12ac + 21ad) + (8bc + 14bd)

Find the GCF of each group.

3a (4c + 7d) + 2b(4c + 7d)

The parentheses are the same!

(3a + 2b)(4c + 7d)

2. Factor rx + 2ry + kx + 2ky

Check for a GCF: None

You have 4 terms - try factoring by grouping.

(rx + 2ry) + (kx + 2ky)

Find the GCF of each group.

r(x + 2y) + k(x + 2y)

The parentheses are the same!

(r + k)(x + 2y)

3. Factor 2x2 - 3xz - 2xy + 3yzCheck for a GCF: None

Factor by grouping. Keep a + between the groups.

(2x2 - 3xz) + (- 2xy + 3yz)

Find the GCF of each group.

x(2x - 3x) + y(- 2x + 3z)

The signs are opposite in the parentheses!

Keep-change-change!

x(2x - 3x) - y(2x - 3z)

(x - y)(2x - 3z)

4. Factor 16k3 - 4k2p2 - 28kp + 7p3

Check for a GCF: None

Factor by grouping. Keep a + between the groups.

(16k3 - 4k2p2 ) + (-28kp + 7p3)

Find the GCF of each group.

4k2(4k - p2) + 7p(-4k + p2)

The signs are opposite in the parentheses!

Keep-change-change!

4k2(4k - p2) - 7p(4k - p2)

(4k2 - 7p)(4k - p2)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

y2 + 6y + 8

y +2

y

+4

y2

+4y

+2y

+8

y2

+4y+2y+8

Review: (y + 2)(y + 4)

In this lesson, we will begin with y2 + 6y + 8 as our problem and finish with (y + 2)(y + 4) as our answer.

Here we go! 1) Factor y2 + 6y + 8Use your factoring chart.

Do we have a GCF?

Is it a Diff. of Squares problem?

Now we will learn Trinomials! You will set up a table with the following information.

Nope!No way! 3 terms!

Product of the first and last coefficients

Middlecoefficient

The goal is to find two factors in the first column that add up to the middle term in the second column.

We’ll work it out in the next few slides.

1) Factor y2 + 6y + 8Create your MAMA table.

Multiply Add+8 +6

Product of the first and last coefficients

Middlecoefficient

Here’s your task…What numbers multiply to +8 and add to +6? If you cannot figure it out right away, write

the combinations.

M

A

1) Factor y2 + 6y + 8Place the factors in the table.

+1, +8

-1, -8

+2, +4

-2, -4

Multiply Add+8 +6

Which has a sum of +6?

+9, NO

-9, NO

+6, YES!!

-6, NO

We are going to use these numbers in the next step!

1) Factor y2 + 6y + 8

+2, +4

Multiply Add+8 +6

+6, YES!!Hang with me now! Replace the middle number of the trinomial with our working numbers from the

MAMA table y2 + 6y + 8

y2 + 2y + 4y + 8Now, group the first two terms and the last two

terms.

We have two groups!(y2 + 2y)(+4y + 8)

If things are done right, the parentheses should be the same.

Almost done! Find the GCF of each group and factor it out.

y(y + 2) +4(y + 2)

(y + 4)(y + 2)

Tadaaa! There’s your answer…(y + 4)(y + 2)You can check it by multiplying. Piece of cake, huh?

There is a shortcut for some problems too! (I’m not showing you that yet…)

Factor out the GCF’s. Write them in their own group.

2) Factor x2 – 2x – 63Create your MAMA table.

Multiply Add-63 -2

Product of the first and last coefficients

Middlecoefficient

-63, 1

-1, 63

-21, 3

-3, 21

-9, 7

-7, 9

-62

62

-18

18

-2

2

Signs need to be different

since number is negative.

M

A

Replace the middle term with our working numbers.

x2 – 2x – 63x2 – 9x + 7x – 63

Group the terms.

(x2 – 9x) (+ 7x – 63)

Factor out the GCF

x(x – 9) +7(x – 9)

The parentheses are the same! Weeedoggie!

(x + 7)(x – 9)

Here are some hints to help you choose your factors in the

MAMA table.

1) When the last term is positive, the factors will have the same sign as the middle term.

2) When the last term is negative, the factors will have different signs.

2) Factor 5x2 - 17x + 14 Create your MAMA table.

Multiply Add+70 -17

Product of the first and last coefficients

Middlecoefficient

-1, -70

-2, -35

-7, -10

-71

-37

-17

Signs need to be the same as

the middle sign since the

product is positive. Replace the middle term.

5x2 – 7x – 10x + 14

Group the terms.

M

A

(5x2 – 7x) (– 10x + 14)Factor out the GCF

x(5x – 7) -2(5x – 7)

The parentheses are the same! Weeedoggie!

(x – 2)(5x – 7)

Hopefully, these will continue to get easier the more you do them.

Factor x2 + 3x + 21. (x + 2)(x + 1)

2. (x – 2)(x + 1)

3. (x + 2)(x – 1)

4. (x – 2)(x – 1)

Factor 2x2 + 9x + 101. (2x + 10)(x + 1)

2. (2x + 5)(x + 2)

3. (2x + 2)(x + 5)

4. (2x + 1)(x + 10)

Factor 6y2 – 13y – 51. (6y2 – 15y)(+2y – 5)

2. (2y – 1)(3y – 5)

3. (2y + 1)(3y – 5)

4. (2y – 5)(3y + 1)

2) Factor 2x2 - 14x + 12

Multiply Add+6 -7

Find the GCF!

2(x2 – 7x + 6)

Now do the MAMA table!

-7

-5

Signs need to be the same as

the middle sign since the

product is positive.

Replace the middle term.

2[x2 – x – 6x + 6]

Group the terms.

-1, -6

-2, -3

2[(x2 – x)(– 6x + 6)]Factor out the GCF

2[x(x – 1) -6(x – 1)]

The parentheses are the same! Weeedoggie!

2(x – 6)(x – 1)

Don’t forget to follow your factoring chart when doing these problems. Always look for a GCF

first!!