5.5 Add and Subtract Rational Expressions

p. 336

What must be true before you can add or subtract complex fractions?

What are two methods to simplify a complex fraction?

Remember: When adding or subtracting fractions, you need a

common denominator!

5

1

5

3 . a

5

4

2

1

3

2 . b

6

3

6

4

6

1

43

21

.c3

4

2

1

6

4

3

2

Examples:

xxa

2

7

2

3 .

x2

4

x

2

4

6

4

3 .

xx

xb

4

63

x

x4

)2(3

x

xor

The Least Common Multiple (LCM) must be made up of all of the factors of each polynomial, BUT - NO REPEATS

2 2 3Answer

Example:

343 363

4

xx

x

x ** Needs a common denominator 1st!

Sometimes it helps to factor the denominators to make it easier to find

your LCD.)12(33

433

xx

x

xLCD: 3x3(2x+1)

)12(3)12(3

)12(433

xx

x

xx

x

)12(3

)12(43

xx

xx)12(3

493

xx

x

ExampleExample:

9

1

96

122

xxx

x

)3)(3(

1

)3)(3(

1

xxxx

x

)3()3(

)3(

)3()3(

)3)(1(22

xx

x

xx

xx

LCD: (x+3)2(x-3)

)3()3(

)3()3)(1(2

xx

xxx

)3()3(

3332

2

xx

xxxx

)3()3(

632

2

xx

xx

Complex Fraction – a fraction with a fraction in the numerator and/or

denominator.

Such as:

How would you simplify this complex fraction?

Multiply the top by the reciprocal of the bottom!

52

31

6

5

2

5*

3

1

Method 1: Steps to make complex fractions easier

1. Condense the numerator and denominator into one fraction each. (if necessary)

2. Multiply the numerator by the inverse of the denominator.

3. Simplify the remaining fraction.

Method 2: Steps to make complex fractions easier

• Multiply the numerator and the denominator by the least common denominator (LCD) of every fraction in the numerator and denominator.

• Simplify

Method 1: 1 fraction numerator, 1 Method 1: 1 fraction numerator, 1 fraction denominatorfraction denominator

13

41

43

xx

x

)1)(4()4(3

)1)(4()1(

)4(3

xxx

xxx

x

)1)(4(1231

)4(3

xxxx

x

)1)(4(114

)4(3

xxxx

114

)1)(4(

)4(

3

x

xx

x

)114)(4(

)1)(4(3

xx

xx

114

33or

114

)1(3

x

x

x

x

ExampleMethod 1:

xx

x1

14

12

)1()1(

)1(4

12

xxx

xxxx

)1(14

12

xxxx

x

)1(151

2

xxxx

15

)1(

1

2

x

xx

x )15)(1(

)1(2

xx

xx

15

2

x

x

Method 2: Find LCD Method 2: Find LCD of num. & den.of num. & den.

Simplify:

The LCD of all the fractions in the numerator and denominator is x(x + 4).

5x + 4

1x + 4+ 2

x

5x + 4

1x + 4+ 2

x

=Multiply numerator and denominator by the LCD.

x + 2(x + 4)5x= Simplify.5x

3x + 8=

5x + 4

1x + 4+ 2

x

Method 2

x6

x3

–

x5

710

–

– 5x 3 (2x – 7)=

Multiply numberator and denominator by the LCD (6,3,5 & 10 all factors of 30)

Simplify

x6

x3

–

x5

710

–=

• What must be true before you can add or subtract complex fractions?

The fractions must have common denominators.

• What are two methods to simplify a complex fraction?

Treat the complex fraction as two problems and find the common denominator of each before you invert and multiply or multiply by the LCD of the numerator and denominator.

5.5 Assignment

p. 340,

3-35 odd