Chapter 7 Section 2 - hammelscienceandmath.weebly.comhammelscienceandmath.weebly.com/uploads/5/7/7/9/57791583/7-2_… · Step 2 Find the least common denominator. The LCD is the product

Section 2 Chapter 7

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

1

Objectives

2

3

Adding and Subtracting Rational Expressions

Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational expressions with different denominators.

7.2


Add and subtract rational expressions with the same denominator.

Objective 1

Slide 7.2- 3


Adding or Subtracting Rational Expressions

Step 1 If the denominators are the same, add or subtract the numerators. Place the result over the common denominator.

If the denominators are different, first find the least common denominator. Write all rational expressions with this least common denominator, and then add or subtract the numerators. Place the result over the common denominator.

Step 2 Simplify. Write all answers in lowest terms.

Slide 7.2- 4

Add and subtract rational expressions with the same denominator.


Add or subtract as indicated.

79 9x y+

79x y+

=

3 3

8 143 3x x

− 3

8 143x−

= 3

63x−

= 3

2x−

=

2 2 2 2

r tr t r t

+− − 2 2

r tr t+

=− ( )( )

r tr t r t

+=

+ −

1r t

=−

2 2

63 18 3 18

xx x x x

++ − + − 2

63 18x

x x+

=+ −

( )( )66 3x

x x+

=+ −

13x

=−

Slide 7.2- 5

CLASSROOM EXAMPLE 1 Adding and Subtracting Rational Expressions Same Denominator

Solution:


Find a least common denominator.

Objective 2

Slide 7.2- 6


Finding the Least Common Denominator

Step 1 Factor each denominator. Step 2 Find the least common denominator. The LCD is the

product of all of the different factors from each denominator, with each factor raised to the greatest power that occurs in any denominator.

Slide 7.2- 7

Find a least common denominator.


Find the LCD for each group of denominators. 3 5 2 610 ,15a b a b

z, z + 6

Factor. 10a3b5 = 2 • 5 • a3 • b5

15a2b6 = 3 • 5 • a2 • b6

LCD = 2 • 3 • 5 • a3 • b6 = 30a3b6

Each denominator is already factored.

LCD = z(z + 6)

Slide 7.2- 8

CLASSROOM EXAMPLE 2 Finding Least Common Denominators

Solution:


Find the LCD for each group of denominators. m2 – 16 , m2 + 8m + 16

Factor. = (m +4)(m – 4)

LCD

m2 – 16

m2 + 8m + 16 = (m +4)2

= (m +4)2(m – 4)

x2 – 2x + 1, x2 – 4x + 3, 4x – 4

x2 – 2x + 1 = (x – 1)(x – 1)

x2 – 4x + 3 = (x – 1)(x – 3)

4x – 4 = 4(x – 1)

LCD = 4(x – 1)2(x – 3) Slide 7.2- 9

CLASSROOM EXAMPLE 2 Finding Least Common Denominators (cont’d)

Solution:


Add and subtract rational expressions with different denominators.

Objective 3

Slide 7.2- 10


Add or subtract as indicated.

6 14m m

+

2 14y y

−+

6 14

44m m⋅

= +⋅

24 14m+

=254m

=

( )( )

( )( )

44

2 14y y yyyy

= −+

+

+

( )( ) ( )2 4

4 4y y

y y y y+

= −+ +

( )2 8

4y yy y+ −

=+

Slide 7.2- 11

CLASSROOM EXAMPLE 3 Adding and Subtracting Rational Expressions (Different Denominators)

Solution:

24 14 4m m

= +

( )84

yy y

+=

+


Subtract.

5 7 142 7 2 7x xx x+ − −

−+ +

The denominators are already the same for both rational expressions. The subtraction sign must be applied to both terms in the numerator of the second rational expression.

15 72 7( 4)xxx

−=

− −+

+5 7 1

74

2x x

x+ ++

=+

6 212 7xx+

=+

( )2 732 7xx+

=+

3=

Slide 7.2- 12

CLASSROOM EXAMPLE 4 Subtracting Rational Expressions

Solution:


2 32 1r

r r+

−− −

The LCD is (r – 2)(r – 1).

( )( )( )

( )( )( )( )1

221

1 232

rr

r rr rr− −

−

+=

− − −−

( )( )( )

22 22 1

6r rrr r−

=−

− +

−

−

( )( )

22 22 1

6r rrr r−

=−

− −

−

+

( )( )

2 42 1r rr r− + +

=− −

Slide 7.2- 13

CLASSROOM EXAMPLE 4 Subtracting Rational Expressions (cont’d)

Subtract.

Solution:


Add.

2 1 .3 3x x+

− −

To get a common denominator of x – 3, multiply both the numerator and denominator of the second expression by –1.

( )( )( )1 12

3 3 1x x−

= +− − −

2 13 3x x

−= +

− −

( )2 13x

+ −=

−

13x

=−

Slide 7.2- 14

CLASSROOM EXAMPLE 5 Adding and Subtracting Rational Expressions (Denominators Are Opposites)

Solution:


Add and subtract as indicated.

2

4 2 105 5x x x x

−+ −

− −

( )4 2 105 5x x x x

−= + −

− −

( )( )( ) ( )

524 15 55

0xxxx x xx x

−= −

−

−+

− −

( )( )( )

4 2 5 105

x xx x

+ − − −=

−

( )4 2 10 10

5x xx x

+ − + −=

− ( )25x

x x=

−

25x

=−

Slide 7.2- 15

CLASSROOM EXAMPLE 6 Adding and Subtracting Three Rational Expressions

Solution:


Subtract.

2 2

43 4 7 12a a

a a a a−

−+ − + +

( )( ) ( )( )4

4 1 4 3a a

a a a a−

= −+ − + +

( )( )( )( )

34 1 3a a

a a a− +

=+ − +

LCD is ( )( )( )4 1 3 .a a a+ − +

( )( )( )( )

4 14 1 3a a

a a a−

−+ − +

Slide 7.2- 16

CLASSROOM EXAMPLE 7 Subtracting Rational Expressions

Solution:


( ) ( )( )( )( )

3 4 14 1 3

a a a aa a a− + − −

=+ − +

( )( )( )

2 23 4 44 1 3

a a a aa a a− − − +

=+ − +

( )( )( )

254 1 3

a aa a a

− +=

+ − +

Distributive property

Combine terms in the numerator.

Slide 7.2- 17

CLASSROOM EXAMPLE 7 Subtracting Rational Expressions (cont’d)


Add.

2 2

4 16 9 2 15p p p p

+− + + −

( )( ) ( )

( )( )( )2 2

4 5 1 33 5 5 3p p

p p p p+ −

= +− + + −

LCD is

( )( ) ( )( )4 13 3 5 3p p p p

= +− − + −

( ) ( )23 5 .p p− +

Slide 7.2- 18

CLASSROOM EXAMPLE 8 Adding Rational Expressions

Solution:


( ) ( )( ) ( )2

4 5 1 33 5

p pp p+ + −

=− +

( ) ( )24 20 3

3 5p pp p+ + −

=− +

( ) ( )25 173 5p

p p+

=− +

Distributive property

Combine terms in the numerator.

Slide 7.2- 19

CLASSROOM EXAMPLE 8 Adding Rational Expressions (cont’d)

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Chapter 7 Section 2 - hammelscienceandmath.weebly.comhammelscienceandmath.weebly.com/uploads/5/7/7/9/57791583/7-2_… · Step 2 Find the least common denominator. The LCD is the product