5.3 Adding and Subtracting Rational Expressions BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 Procedure: To add or subtract fractions with.

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5.3 Adding and Subtracting Rational Expressions

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

Procedure: To add or subtract fractions with like denominators.

1. Write a single fraction using the like denominator as the denominator.

2. For the numerator of the single fraction, write each of the numerators with the operation symbol between them.

Note: If the operation is subtraction, and the numerator of the second fraction has more than one term, then write parentheses around around the numerator followingthe subtraction symbol as a reminder for distributing. 3. Simplify the numerator.

9

4. Factor the numerator and denominator if possible and divide out any common factors.

a 3 7Simplify Example 1.

9 9


2. For the numerator of the single fraction, write each of the numerators with the operation symbol between the numerators.

a 3 7

3. Simplify the numerator.a 4Answer:

9

Your Turn Problem #1

3x 2x 1Simplify:

5 5

x 1Answer:

5

2



2 2

3x 12. SimplExampl ifye 2

6x 6x

2

6x

3x 12

3. Factor the numerator.2

3(x 4)6x

4. Divide out like factors.2

2

x 42x


2. For the numerator of the single fraction, write each of the numerators with the operation symbol between the numerators.


6a 18Simplify:

a 3 a 3

Answer: 6

3



Procedure: To add or subtract fractions with unlike denominators.

Adding and Subtracting Fractions with Unlike Denominators

1. Find the LCD of the denominators. (The lowest polynomial that contains the factors of each polynomial.)2. Multiply numerator and denominator by the same factor which makes denominator equal to the LCD (multiply only by what you need).

3. Add or subtract the numerators and combine like terms.

4. Factor the numerator and divide out the common factors if any exist.

3x 5 Simplify Exam

1ple 3.

6 2

1 16 2 2 2 2

12 2 2 3LCD 2 2 2 2 3 48

2. Multiply numerator and denominator of 1st fraction by 3. Multiply numerator and denominator 2nd fraction by 4.

(3) (4)

(3)3x 516 )12(4

9x 20Answer:

48

1. Find LCD.

Your Turn Problem #33a 1

Simplify: 4 6

9a 2

Answer: 12

3. Simplify.

4



9 3 36 2 3LCD 2 3 3 18

4x 2 15x 618

3. Add the numerators and combine like terms.

11x 4Answer :

18

4. Nothing to factor from numerator.

2x 1 5x 2 SimplExampl ify

94.

6e

( 2x 1 5x 2

9 6

2) (3)

(2) (3)

1. Find LCD.

2. Multiply numerator and denominator of 1st fraction by 2. Multiply numerator and denominator 2nd fraction by 3. Note: use parentheses to show distribution.


4x 3 8x 12Simplify:

6 12

1Answer:

2

5



2

2

LCD of 3 and 2 is 6.Then take the highest degreeof any variable,Then LCD =

x . 6x

2

10x 12 9x6x

2

5 2 3 SimplExamp ify

3xle

2x.

x5

2

2x 6 3x

2x 65 2 33x 2xx 3x

2

19x 12Answer :

6x

3. Add the numerators and combine like terms.

1. Find LCD.

2. Change each fraction to an equivalent fraction with a denominator of 6x2.


2

7 9 5Simplify:

4x 2x3x

2 2

28 57x 57x 28Answer: or

12x 12x

6



9x 3 4x 6(2x 3)(3x 1)

3. Write as one fraction with the LCD.

Distribute (don’t forget the negative). Then combine like terms.

3 2 Simplify Exa

2xmple 6.

3 3x 1

2. Change each fraction to an equivalent fraction with a denominator of (2x 3)(3x 1).

5x 3Answer :

(2x 3)(3x 1)

The LCD is contains factors from each denominator.LCD =(2x 3)(3x 1).

3x 1 2x 3

3x3 2

2x 3 3 x 3x1 1 2

1. Find LCD.


5 3Simplify:

x 1 2x 3

7x 12Answer:

(x 1)(2x 3)

7



6x 2 4x3x 1

4x Simplify 2Example

17.

3x

2. Change each fraction to an equivalent fraction with a denominator of (3x 1).

10x 2Answer:

3x 1

The LCD is contains factors from each denominatoLCD =

r.(3x 1)

3x 1 1

3x 12 4x1 3 1x 1

1. Find LCD.

3. Write as one fraction with the LCD. Distribute. Then combine like terms.


5Simplify: 2

4x 3

8x 1Answer:

4x 3

8



To add or subtract rational expressions where the denominator is factorable, we will have to add a step to the previous procedure.

Procedure: Adding rational expressions if one or more denominators is factorable:1. Factor each denominator2. Continue the process given in the previous lesson.

2

2x 7 Simplify

x 4x 16Example 8. 1. Factor denominators.

2x 7(x 4)(x 4) x 4

2. Find LCD.LCD (x 4)(x 4)

3. Change each expression to an equivalent fraction with the LCD as the denominator.

2x 7(x 4)(x 4) (

(x

x 4)

4)

(x 4)

4. Combine the fractions and simplify.2x 7x 28

(x 4)(x 4)

9x 28

(x 4)(x 4)


2

4a 4 3Simplify:

a 2a 4

1Answer:

a 2

9



2

3x 2 SimplifExamp y

5x 30x 36le 9. 1. Factor denominators

3x 2(x 6)(x 6) 5(x 6)

2. Find LCDLCD 5(x 6)(x 6)


(5) (x 6)

(3x 2

(x 6)(x 6) 5(x 6)5) (x 6)

4. Combine the fractions and simplify. (Don’t forget to distribute the negative.)

15x 2x 125(x 6)(x 6)

13x 12Answer:

5(x 6)(x 6)


2

2b 4Simplify:

3b 9b 9

2b 12Answer:

3(b 3)(b 3)

10



2

5 1 2 Simplify

x 2 xExamp

xle 10.

2x1. Factor denominators

2. Find LCDLCD x(x 2)

4. Combine the fractions and simplify.

5x x 2 2x(x 2)

6x

x(x 2)

6Answer:

(x 2)


5 1 2x 2 x x(x 2)

5 1 2x 2 x

(x) (x 2)

(x) ( x 2x x2) ( )


2

5x 2 6Simplify:

x 3 x x 3x

5x 2Answer:

x 3

11



2

4x 2 4 SimplifExample y

x 1 x 11 .

x 1

11. Factor denominators

2. Find LCDLCD (x 1)(x 1)

4. Combine the fractions and simplify.

24x 4x 2x 2 4(x 1)(x 1)

2(2x 1)

(x 1)

4x 2 4x 1 x 1 (x 1)(x 1)

3. Change each expression

to an equivalent fraction with the LCD as the denominator.

(x 1) (x 1)

(x 1) (4x 2 4

x 1 xx 1 1 (x 1)(x 1) )

24x 2x 2(x 1)(x 1)

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


2

3 x 5 3Simplify:

x 1 x 1x 1

1Answer:

x 1

12



2 2

6 10 Simplify

a 3a 54Example 1

a.

5a2

6 1. Factor denominators

2. Find LCDLCD (a 9)(a 6)(a 1)

4. Combine the fractions and simplify.6a 6 10a 90

(a 9)(a 6)(a 1)

4a 84Answer:

(a 9)(a 6)(a 1)


6 10(a 9)(a 6) (a 6)(a 1)

6 10(a 9)(a 6) (a 6)(a

(a 1) (a 9)

(a 1) ( 9)1) a


2 2

3 2Simplify:

x x 6 x 2x 3

x 1Answer:

(x 3)(x 2)(x 1)

5.3 Adding and Subtracting Rational Expressions BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 Procedure: To add or subtract fractions with.

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