Monday, June Monday, June 3030

FactoringFactoring

Factoring out the GCF

Greatest Common FactorGreatest Common Factor

The greatest common factor (GCF) is The greatest common factor (GCF) is the product of what both items have the product of what both items have in common.in common.

Example: 18xy , 36y2

18xy = 2 · 3 · 3 · x · y 36y2 = 2 · 2 · 3 · 3 · y · y

GCF =

= 18y2 · 3 · 3 · y

Now you try!Now you try!

Example 1:Example 1: 12a12a22b , 90ab , 90a22bb22cc

Find the greatest common factor of the following:

Example 2:Example 2: 15r15r22 , 35s , 35s22 , 70rs , 70rs

GCF = 6a2b

GCF = 5

FactoringFactoring- Opposite of Opposite of distributingdistributing

- Breaking down a Breaking down a polynomial to what polynomial to what multiplies together to multiplies together to form the polynomialform the polynomial

Example:Example:

Factor:Factor: 12a12a22 + 16a + 16a

= 2·2·3·a·a + 2·2·2·2·a= 2·2·3·a·a + 2·2·2·2·a

= = 22 · · 22 · · aa (3·a + 2·2)

= 4a (3a + 4)

You can check by distributing.

1. Factor each term.

2. Pull out the GCF.

3. Multiply.

Example:Example:

Factor: Factor: 18cd18cd22 + 12c + 12c22d + 9cdd + 9cd

= 2·3·3·c·d·d + 2·2·3·c·c·d + = 2·3·3·c·d·d + 2·2·3·c·c·d + 3·3·c·d3·3·c·d= = 33 · · cc · · dd (2·3·d + 2·2·c + 3)

= 3cd (6d + 4c + 3)

Now you try!Now you try!

Example 1:Example 1:

15x + 25x15x + 25x22

Example 2:Example 2:

12xy + 24xy12xy + 24xy22 – 30x – 30x22yy44

= 6xy(2 + 4y – 5xy3)

= 5x(3 + 5x)

Factoring by Factoring by GroupingGrouping

Example:Example:Factor:Factor: 5xy – 35x + 3y – 21 5xy – 35x + 3y – 21

(5xy – 35x)(5xy – 35x) + + (3y – 21)(3y – 21)

= (5·x·y – 5·7·x)

+ (3·y – 3·7)

= 5·x (y – 7)+ 3 (y – 7)

= 5x (y – 7)+ 3 (y – 7)

= (5x + 3)(y – 7)

Example:Example:Factor:Factor: 5xy – 35x + 3y – 21 5xy – 35x + 3y – 21

(5xy – 35x)(5xy – 35x) + + (3y – 21)(3y – 21)

= 5x (y – 7)+ 3 (y – 7)

= (5x + 3)(y – 7)

1. Group terms with ( ).

2. Pull out GCF from each group.3. Split

into factors.

NotesNotes- What is in parentheses - What is in parentheses MUST be the same!!MUST be the same!!

- Grouping only works - Grouping only works if there are 4 terms!!if there are 4 terms!!

Now you try!Now you try!

Factor.Factor.

Example 1:Example 1: 5y5y22 – 15y + 4y - 12 – 15y + 4y - 12

Example 2:Example 2: 5c – 10c5c – 10c22 + 2d – 4cd + 2d – 4cd

= (5y + 4)(y – 3)

= (5c + 2d)(1 – 2c)

2 more important examples:2 more important examples:

Example 1:Example 1: 2xy + 7x + 2y + 72xy + 7x + 2y + 7

(2xy + 7x) + (2y + 7)

+ (2y + 7)= x (2y + 7)

= (x + 1)

+ 1(2y + 7)

(2y + 7)

Example 2:Example 2: 15a – 3ab – 20 + 4b15a – 3ab – 20 + 4b

(15a – 3ab) – (20 + 4b)

– 4 (5 – b)= 3a (5 – b)

= (3a – 4)(5 – b)

–(15a – 3ab) – (20 – 4b)

If there is a negative in the middle, you MUST change the sign after it.

Factoring Factoring TrinomialsTrinomials

Example 1:Example 1:

Factor: xFactor: x22 + 5x + 6 + 5x + 6 66

1 · 61 · 62 · 32 · 3

Look for factors of 6 thatADDto positive

5

(x + 2)

(x + 3)

Example 2:Example 2:

Factor: xFactor: x22 + 7x + 12 + 7x + 12 1212

1 · 1 · 12122 · 62 · 6

Look for factors of 12 that

ADDto positive

7

(x + 3)

(x + 4) 3 · 43 · 4

Now you try!Now you try!

Example: xExample: x22 + 6x + 8 + 6x + 8

Example: xExample: x22 + 11x + 10 + 11x + 10

(x + 2)(x + 4)

(x + 1)(x + 10)

To determine the signs:To determine the signs:

Last sign

Positive Negative( + )( – )Middle

sign

Positive Negative( + )

( + )( – )( – )

Example 3:Example 3:

Factor: xFactor: x22 – 12x + 27 – 12x + 27 2727

1 · 1 · 27273 · 93 · 9

Look for factors of 27 that

ADDtonegative

12

(x – 3)(x – 9)

Example 4:Example 4:

Factor: xFactor: x22 + 3x – 18 + 3x – 18 1818

1 · 1 · 18182 · 92 · 9

Look for factors of 18 that

SUBTRACTtopositive 3

(x + 6)

(x – 3) 3 · 63 · 6

Now you try!Now you try!

Example: xExample: x22 – x – 20 – x – 20

Example: xExample: x22 – 7x – 18 – 7x – 18

(x + 4)(x – 5)

(x + 2)(x – 9)

Please note!Please note!

Example: xExample: x22 – 5x – 6 – 5x – 6

Example: xExample: x22 – 5x + 6 – 5x + 6

(x + 1)(x – 6)

(x – 2)(x – 3)

More Factoring More Factoring TrinomialsTrinomials

Example 1:Example 1:

Factor: 6xFactor: 6x22 + 17x + 5 + 17x + 5 3030

1 · 1 · 30302 · 2 · 15153 · 3 · 10105 · 65 · 6

6x2 + 2x + 15x + 5

(6x2 + 2x) + (15x + 5)

2x(3x + 1) + 5(3x + 1)(2x + 5)(3x + 1)

Example 2:Example 2:

Factor: 4xFactor: 4x22 + 24x + 32 + 24x + 32

Always check your factors to see if there is anything more that can be factored out.

OR Example 2:OR Example 2:

Factor: 4xFactor: 4x22 + 24x + 32 + 24x + 32

It is usually faster if you factor out the GCF first.

Always check to see if there is anything you can factor out first.

Now you try!Now you try!

Example: 5xExample: 5x22 + 27x + 10 + 27x + 10

Example: 24xExample: 24x22 – 22x + 3 – 22x + 3

(5x + 2)(x + 5)

(4x – 3)(6x – 1)