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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)