copyright © 2013, 2009, 2006 pearson education, inc. 1 section 6.1 the greatest common factor and...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 6.1

The GreatestCommon Factorand Factoringby Grouping



Factoring

23 6a a

Factoring a polynomial means finding an equivalent expression that is a product. For example, when we take the polynomial

3 2a a and write it as

we say that we have factored the polynomial. In factoring, we write asum as a product.


Objective #1 Find the greatest common factor.


Greatest Common Factor - GCF

• The greatest common factor is an expression of the highest degree that divides each term of the polynomial.

• The variable part of the greatest common factor always contains the smallest power of a variable that appears in all terms of the polynomial.

Factoring


Objective #1: Example

1a. Find the greatest common factor of the following list of monomials: 318x and 215x

3 2

2 2

18 3 6

15 3 5

x x x

x x

The GCF is 23 .x



1a. Find the greatest common factor of the following list of monomials: 318x and 215x

3 2

2 2

18 3 6

15 3 5

x x x

x x

The GCF is 23 .x



1b. Find the greatest common factor of the following list of

monomials: 4 3 2, ,x y x y and 2x y

4 2 2

3 2 2

2 2

x y x y x

x y x y xy

x y x y

The GCF is 2 .x y


Objective #2 Factor out the greatest common factor of a

polynomial.


Factoring a Monomial from a Polynomial

1. Determine the greatest common factor of all terms in the polynomial.

2. Express each term as the product of the GCF and its other factor.

3. Use the distributive property to factor out the GCF.

Factoring


Factor: 7x2 + 28x

The GCF of 7x2 and 28x is 7x.

7x2 + 28x = 7x·x +7x·4

= 7x(x + 4)

Express each term as the product of the GCF and the its other factor.

Factor out the GCF.

Determine the GCF.

Greatest Common Factor

EXAMPLEEXAMPLE


Factor: 7x2 + 28x

The GCF of 7x2 and 28x is 7x.

7x2 + 28x = 7x·x +7x·4

= 7x(x + 4)


Factor out the GCF.

Determine the GCF.


EXAMPLEEXAMPLE


Factor: 5x4y − 10x2y2 + 25xy

The GCF of 5x4y, – 10x2y2 , and

25xy is 5xy.

5x4y – 10x2y2 + 25xy

= 5xy·x3 − 5xy·2xy + 5xy·5

= 5xy(x3 – 2xy + 5)


Factor out the GCF.

Determine the GCF.


EXAMPLEEXAMPLE



2a. Factor: 26 18x

The GCF is 6.

2 2

2

6 18 6 6 3

6( 3)

x x

x


2b. Factor: 2 325 35x x

The GCF is 25 .x

2 3 2 2

2

25 35 5 5 5 7

5 (5 7 )

x x x x x

x x




2c. Factor: 5 4 315 12 27x x x

The GCF is 33 .x

5 4 3 3 2 3 3

3 2

15 12 27 3 5 3 4 3 9

3 (5 4 9)

x x x x x x x x

x x x



2d. Factor: 3 2 28 14 2x y x y xy

The GCF is 2 .xy

3 2 2 2

2

8 14 2 2 4 2 7 2 1

2 (4 7 1)

x y x y xy xy x y xy x xy

xy x y x


Objective #3 Factor out the negative of the greatest

common factor of a polynomial.


Factoring

When factoring polynomials, it is preferable to have a first term with a positive coefficient inside parentheses. We can do this with by factoring out −5, the negative of the GCF.

Express each term as the product of the negative of the GCF and its other factor. Then use the distributive property to factor out the negative of the GCF.

25 30x

2

2

2

5 30

5 5 6

5 6

x

x

x


Factoring

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

xxx 10505 23

251055 2 xxxxx

3 25 50 10 x x x Factor:

The GCF is 5x. Because the leading coefficient, −5, is negative, we factor out a common factor with a negative coefficient. We will factor out the negative of the GCF, or −5x.

Factor out the GCF 2105 2 xxx

Express each term as the product of the GCF and its other factor



3. Factor: 4 5 3 4 216 24 20a b a b ab

It is preferable to have a first term with a positive coefficient inside parentheses.

Thus, we will factor out 24 .ab

4 5 3 4 2

2 3 3 2 2 2 2

2 3 3 2 2

16 24 20

4 4 4 ( 6 ) 4 5

4 (4 6 5)

a b a b ab

ab a b ab a b ab

ab a b a b


Objective #4 Factor by grouping.


Factoring by Grouping

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

yxyxx 7

. 7 yxyxx Factor:

Identify the common binomial factor in each part of the problem.

The GCF, a binomial, is x + y.

The GCF, a binomial, is x + y.



yxyxx 17

17 xyx

yxyxx 7

Factor out the common binomial factor as follows.

Factor out the GCF

This step, usually omitted, shows each term as the product of the GCF and its other factor, in that order.

CONTINUEDCONTINUED



Factoring by Grouping1) Group terms that have a common monomial factor. There will usually be two groups. Sometimes the terms must be rearranged.

2) Factor out the common monomial factor from each group.

3) Factor out the remaining common binomial factor (if one exists).


Factor: 3x2 + 6x + 4x + 8 by grouping.

3x2 + 6x + 4x + 8

= (3x2 + 6x) + (4x + 8)

= 3x(x + 2) + 4(x + 2)

= (x + 2)(3x + 4)

Group terms that have a common monomial factor.

Factor out the common monomial factor from each group.

Factor out the remaining common binomial factor.


EXAMPLEEXAMPLE


Factor: 3x2 + 6x + 4x + 8 by grouping.

3x2 + 6x + 4x + 8

= (3x2 + 6x) + (4x + 8)

= 3x(x + 2) + 4(x + 2)

= (x + 2)(3x + 4)





EXAMPLEEXAMPLE


Factor: 5x2 – xy + 15xy – 3y2

5x2 – xy + 15xy – 3y2

= (5x2 – xy) + (15xy – 3y2)

= x(5x – y) + 3y(5x – y)

= (5x – y)(x + 3y)





EXAMPLEEXAMPLE



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

2 23 6 2 4a x a y bx by

yaxa 22 63

Factor:

There is no factor other than 1 common to all terms. However, we can group terms that have a common factor:

Common factor is :

bybx 42

yxayaxa 2363 222

+

23a



bybxyaxa 4263 22

bybxyaxa 4263 22

It is sometimes necessary to use a factor with a negative coefficient to obtain a common binomial factor for the two groupings. We now factor the given polynomial as follows:

Group terms with common factors

bayx 232 2

yxbyxa 2223 2 Factor out the common factors from the grouped terms

Factor out the GCF

CONTINUEDCONTINUED



4a. Factor: 2 ( 1) 7( 1)x x x

Factor out the greatest common factor of 1.x GCF GCF

2 2( 1) 7 ( 1) ( 1)( 7)x x x x x



4b. Factor: ( 4) 7( 4)x y y Factor out the greatest common factor of 4.y

GCF GCF

( 4) 7 ( 4) ( 4)( 7)x y y y x



4c. Factor: 3 25 2 10x x x

There is no factor other than 1 common to all four terms. However, we can group terms that have a common factor.

3 2 3 25 2 10 ( 5 ) (2 10)x x x x x x Factor out the greatest common factor from the grouped terms. The remaining two terms have 5x as a common binomial factor, which should then be factored out.

3 2 3 2

2

2

5 2 10 ( 5 ) (2 10)

( 5) 2( 5)

( 5)( 2)

x x x x x x

x x x

x x



4c. Factor: 3 25 2 10x x x


3 2 3 25 2 10 ( 5 ) (2 10)x x x x x x Factor out the greatest common factor from the grouped terms. The remaining two terms have 5x as a common binomial factor, which should then be factored out.

3 2 3 2

2

2

5 2 10 ( 5 ) (2 10)

( 5) 2( 5)

( 5)( 2)

x x x x x x

x x x

x x



4d. Factor: 3 5 15xy x y


3 5 15 ( 3 ) ( 5 15)xy x y xy x y Factor out x from the first two grouped terms and 5 from the last two grouped terms.

3 5 15 ( 3) 5( 3)xy x y x y y Finally, factor out the common factor of 3.y

3 5 15 ( 3) 5( 3)

( 3)( 5)

xy x y x y y

y x

copyright © 2013, 2009, 2006 pearson education, inc. 1 section 6.1 the greatest common factor and...

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