§ 6.2 Adding and Subtracting Rational Expressions

§ 6.2

Adding and Subtracting Rational Expressions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.2

Adding Rational Expressions

In this section, you will practice adding and subtracting rational expressions. Rememberthat when adding or subtracting fractions, it is necessary to rewrite the fractionsas fractions having the same denominator, which is called the common denominator for the fractions being combined.



Adding Rational Expressions With Common Denominators

If are rational expressions, then

To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, simplify the result.

R

Q

R

P and

R

QP

R

Q

R

P



EXAMPLEEXAMPLE

Add: .33

22

2

2

2

xx

xx

xx

xx

SOLUTIONSOLUTION

Add numerators. Place this sum over the common denominator.

Factor.

This is the original expression.xx

xx

xx

xx

33

22

2

2

2

xx

xxxx

3

22

22

xx

xx

3

22

2

3

12

xx

xx

Combine like terms.



Simplify.

Factor and simplify by dividing out the common factor, x.

3

12

xx

xx

CONTINUECONTINUEDD

3

12

x

x



Check Point 1Check Point 1

Add:

SOLUTIONSOLUTION

Add numerators. Place this sum over the common denominator.

Factor.

Combine like terms.

Simplify.


Subtracting Rational Expressions

Subtracting Rational Expressions With Common Denominators

If are rational expressions, then

To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, simplify the result.

R

Q

R

P and

R

QP

R

Q

R

P



EXAMPLEEXAMPLE

Subtract: .6

12

6

2622

2

xx

x

xx

xx

SOLUTIONSOLUTION

Subtract numerators. Place this difference over the common denominator.

This is the original expression.

Remove the parentheses and distribute.

6

12

6

2622

2

xx

x

xx

xx

6

12262

2

xx

xxx

6

12262

2

xx

xxx



Factor.

Combine like terms.

Factor and simplify by dividing out the common factor, x + 3.

6

342

2

xx

xx

CONTINUECONTINUEDD

23

31

xx

xx

23

31

xx

xx

2

1

x

xSimplify.



Check Point 2Check Point 2

Subtract:

SOLUTIONSOLUTION

Subtract numerators. Place difference over the common denominator.

Factor.

Combine like terms.

Simplify.


Least Common Denominators

Finding the Least Common Denominator (LCD)

1) Factor each denominator completely.

2) List the factors of the first denominator.

3) Add to the list in step 2 any factors of the second denominator that do not appear in the list.

4) Form the product of each different factor from the list in step 3. This product is the least common denominator.



EXAMPLEEXAMPLE

Find the LCD of: .472

and 20

322 yy

y

yy

SOLUTIONSOLUTION


45202 yyyy

412472 2 yyyy


4 ,5 yy



3) Add any unlisted factors from the second denominator. The second denominator is (2y - 1)(y + 4). One factor of y + 4 is already in our list, but the factor 2y – 1 is not. We add the factor 2y – 1 to our list.

4) The least common denominator is the product of all factors in the final list. Thus,

12 ,4 ,5 yyy

CONTINUECONTINUEDD

1245 yyy

is the least common denominator.



Check Point Check Point 33

Find the LCD of: .9

2 and

6

72 xx

SOLUTIONSOLUTION



3) Add any unlisted factors from the second denominator.

4) The least common denominator is the product of all factors in the final list.




Find the LCD of: .96

9 and

155

722 xxxx

SOLUTIONSOLUTION



3) Add any unlisted factors from the second denominator.

4) The least common denominator is the product of all factors in the final list.


Add & Subtract Fractions

Adding and Subtracting Rational Expressions That Have Different Denominators

1) Find the LCD of the rational expressions.

2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.

3) Add or subtract numerators, placing the resulting expression over the LCD.

4) If possible, simplify the resulting rational expressions.


Adding Fractions

EXAMPLEEXAMPLE

Add: .23

3

82

722

xxxx

x

SOLUTIONSOLUTION

1) Find the least common denominator. Begin by factoring the denominators.

24822 xxxx

21232 xxxx

The factors of the first denominator are x + 4 and x – 2. The only factor from the second denominator that is unlisted is x – 1. Thus, the least common denominator is,

. 124 xxx


Adding Fractions

2) Write equivalent expressions with the LCD as denominators.


CONTINUECONTINUEDD

23

3

82

722

xxxx

x

21

3

24

7

xxxx

xFactored denominators.

421

43

124

17

xxx

x

xxx

xx Multiply each numerator and denominator by the extra factor required to form the LCD.

. 124 xxx


Adding Fractions

3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible.

Add numerators.

CONTINUECONTINUEDD

Perform the multiplications using the distributive property.

124

4317

xxx

xxx

124

12377 2

xxx

xxx

124

1247 2

xxx

xxCombine like terms.


Adding Fractions

CONTINUECONTINUEDD

Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,

.124

1247 2

xxx

xx


Adding Fractions


Add

SOLUTIONSOLUTIONx

xx

33

32

Multiply

Like fractions

Simplified

233 xx


Adding Fractions


Add

SOLUTIONSOLUTION4

4

x

x

Multiply

Like fractions

Simplify

Combine like terms


Subtracting Fractions

EXAMPLEEXAMPLE

Subtract: .65

3

67

1222

xx

x

xx

x

SOLUTIONSOLUTION


16672 xxxx

61652 xxxx

The factors of the first denominator are x - 4 and x – 1. The only factor from the second denominator that is unlisted is x + 1. Thus, the least common denominator is,

. 116 xxx



2) Write equivalent expressions with the LCD as denominators.


CONTINUECONTINUEDD

Factored denominators.

Multiply each numerator and denominator by the extra factor required to form the LCD.

65

3

67

1222

xx

x

xx

x

61

3

16

12

xx

x

xx

x

161

13

116

112

xxx

xx

xxx

xx

. 116 xxx



3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible.

Subtract numerators.

CONTINUECONTINUEDD

Perform the multiplications using the distributive property and FOIL.

Remove parentheses.

116

13112

xxx

xxxx

116

33122 22

xxx

xxxxxx

116

33122 22

xxx

xxxxxx



Combine like terms in the numerator.

CONTINUECONTINUEDD

116

42

xxx

xx


.116

42

xxx

xx



Check Point Check Point 77Subtract

SOLUTIONSOLUTION

13

23

xx

xx

Multiply

Like fractions

Combine like terms and Simplify

Subtract (add the opposite)


Adding and Subtracting of Fractions

EXAMPLEEXAMPLE

Perform the indicated operations: .1

4

32

2

3 2

xxx

x

x

x

SOLUTIONSOLUTION


313 xx

31322 xxxx

The factors of the first denominator are 1 and x – 3. The only factor from the second denominator that is unlisted is x + 1. We have already listed all factors from the third denominator. Thus, the least common denominator is,

111 xx



. 13or 131 xxxx

CONTINUECONTINUEDD

2) Write equivalent expressions with the LCD as denominator.

1

4

32

2

3 2

xxx

x

x

x This is the original expression.

1

4

13

2

3

xxx

x

x

x Factor the second denominator.

31

34

13

2

13

1

xx

x

xx

x

xx

xx Multiply each numerator and denominator by the extra factor required to form the LCD.



CONTINUECONTINUEDD 3) & 4) Add and subtract numerators, putting this result

over the LCD. Simplify if possible.

13

3421

xx

xxxx Add and subtract numerators.

13

12422

xx

xxxx Perform the multiplications using the distributive property.

13

1422

xx

xxCombine like terms in the numerator.



CONTINUECONTINUEDD

.13

1422

xx

xx



Addition of Fractions (opposites)

EXAMPLEEXAMPLE

Add: .109

22 xyyx

x

SOLUTIONSOLUTION

xyyx

x

109

22

xyyxyx

x

109

xyyxyx

x

10

1

19

xyyxyx

x

109


Factor the first denominator.

Multiply the numerator and the denominator of the second rational expression by -1.

Multiply by -1.



yxyxyx

x

109

Rewrite –y + x as x – y.

Notice the LCD is (x + y)(x – y).

yxyx

yx

yxyx

x

109 Multiply the second numerator

and denominator by the extra factor required to form the LCD.

yxyx

yx

yxyx

x

10109 Perform the multiplications

using the distributive property.

CONTINUECONTINUEDD



yxyx

yxx

10109

Add and subtract numerators.

yxyx

yxx

10109 Remove parentheses and

distribute.

yxyx

yx

1019 Combine like terms in the

numerator.


.1019

yxyx

yx

CONTINUECONTINUEDD


Addition of Fractions

Important to Important to RememberRemember

Adding or Subtracting Rational Expressions

If the denominators are the same, add or subtract the numerators and place theresult over the common denominator.

If the denominators are different, write all rational expressions with the least common denominator (LCD). Once all rational expressions are written in terms of the LCD, then add or subtract as described above.

In either case, simplify the result, if possible. Even when you have used the LCD, it may be true that the sum of the fractions can be reduced.


Addition of Fractions

Important to Important to RememberRememberFinding the Least Common Denominator (LCD)

The LCD is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.

That is - After factoring the denominators completely, the LCD can be determined by taking each factor to the highest power it appears in any factorization.

The Mathematics Teacher magazine accused the LCD of keeping up with the Joneses. The LCD wants everything (all of the factors) the other denominators have.

DONE

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§ 6.2 Adding and Subtracting Rational Expressions