Complex fractions

Objective

• Simplify complex fractions

• Lets Review fraction rules first…………..

Multiplying Fractions

0d and 0b,db

ca

d

c

b

a

Multiply . - 521

·34

- 521

·34

=- 521

·34

1

7

- 57

·14

= =- 528

Multiplying Rational Expressions

1. Factor all numerators and denominators completely.

2. Divide out common factors.3. Multiply numerators together and

multiply denominators together.

Multiply .yx

22z

11z

y18x-52

32

52

32

yx

22z

11z

y18x-4

2

y

36z

52

32

yx

22z

11z

y18x-

Dividing Two Fractions

0c and 0d , 0b ,bc

ad

c

d

b

a

d

c

b

a

Divide . - 2 9

59

=- 2 9

59

- 2 9

·95

1- 2 5

=

1

- 2 9

·95

=

Dividing Rational Expressions

Invert the divisor (the second fraction) and multiply

Divide .3017-x

1

127xx

122

1

3017-x

187xx

1

3017-x

1

187xx

1 2

222

1

15)2)(x(x

2)9)(x(x

1

9)(x

15)(x

Adding/Subtracting Fractions

0c ,c

ba

c

b

c

a 0c ,

c

ba

c

b

c

a

712

= 512

212

+

Add . 512

212

+

Common Denominators

1. Add or subtract the numerators.2. Place the sum or difference of the

numerators found in step 1 over the common denominator.

3. Simplify the fraction if possible.

Subtract .5

6

5

7-2x

5

13-2x

5

6-7-2x

5

6

5

7-2x

Common Denominators

a.) Add .12ww

4-2w-

12ww

53w22

Example:

12ww

4-2w-53w

12ww

4-2w-

12ww

53w222

1)(w

1

1)(w

1w2

12ww

4-2w-53w2

Common Denominators

b.) Subtract

.649x

29x-x

649x

54x2

2

2

2

649x

29)x-(x-54x

649x

29x-x

649x

54x2

22

2

2

2

2

649x

24x3x

649x

29xx-54x2

2

2

22

8)(3x

3)(x

8)8)(3x(3x

8)3)(3x(x

Example:

Unlike Denominators

1. Determine the LCD.2. Rewrite each fraction as an

equivalent fraction with the LCD.3. Add or subtract the numerators

while maintaining the LCD.4. When possible, factor the

remaining numerator and simplify the fraction.

Unlike Denominators

a.)w

5

2w

3

2w

2w

w

5

w

w

2w

3

The LCD is w(w+2).

2)w(w

2)5(w

2)w(w

3w

2)w(w

105w

2)w(w

3w

2)w(w

108w

answers. acceptable also are and 2ww

108w

2)w(w

5)2(4w2

Example:

Unlike Denominators

b.)3x

1

4-4x

x The LCD is 12x(x – 1).

3x

1

1)-4(x

x

1)-4(x

1)-4(x

3x

1

3x

3x

1)-4(x

x

1)-12x(x

44x3x

1)-12x(x

1)-4(x

1)-12x(x

3x 22

This cannot be factored any further.

Example:

Complex Fractions

Simplifying Complex Fractions

A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.

454

3xx

3x

ba9-a

ba

Example:

So how can we simplify them?

• Remember, fractions are just division problems.• We can rewrite the complex fraction as a division

problem with two fractions.• This division problem then changes to multiplication

by the reciprocal.

5

62

3

5

6

2

3

5

6

3

2

5

4

Simplifying Complex Fractions Rule

• Any complex fraction

dcba

Where b ≠ 0, c ≠ 0, and d ≠ 0, may be expressed as:

bc

ad

What if we have mixed numbers in the complex fraction?

• If we have mixed numbers, we treat it as an addition problem with unlike denominators.

• We want to be working with two fractions, so make sure the numerator is one fraction, and the denominator is one fraction

• Now we can rewrite the complex fraction as a division of two fractions

Example

21

25

Try on your own…

4

11

3

What about complex rational expression?

• Treat the complex rational expression as a division problem

• Add any rational expressions to form rational expressions in the numerator and denominator

• Factor• Simplify• “Bad” values

Ex. 2: Simplify .11

11

yx

yx

xyx

xyy

xyx

xyy

yx

yx

11

11

xyxy

xyxy

← The LCD is xy for both the numerator and the denominator.

← Add to simplify the numerator and subtract to simplify the denominator.

xy

xy

xy

xy

← Multiply the numerator by the reciprocal of the

denominator.

Ex. 2: Simplify .11

11

yx

yx

xy

xy

xy

xy

← Eliminate common factors.

xy

xy

Example

x 1

xx 1

(x 1

x) (x 1)

(x 2 1

x)

1

x 1

(x 1)(x 1)

x

1

x 1

x 1

x, x 0, 1

Example

x 2

1 5

x 6

x 2

Try on your own

2

3x1

x

One more for you

x 16

xx 2 8x 16

Ex. 3: Simplify

348

11

41

4

xx

xx

348)3)(11(

41)4)(4(

xxxxxx ← The LCD of the numerator is x +

4, and the LCD of the denominator is x – 3.

Ex. 3: Simplify

348

11

41

4

xx

xx

348338

41168

2

2

xxxxxx

← FOIL the top and don’t forget to subtract the 1 and add the 48 on the bottom.

Ex. 3: Simplify

348

11

41

4

xx

xx

3158

4158

2

2

xxx

xxx

← Simplify by subtracting the 1 in the numerator and adding the 48 in the denominator.

Ex. 3: Simplify

348

11

41

4

xx

xx

158

3

4

1582

2

xx

x

x

xx

← Multiply by the reciprocal.

x2 + 8x +15 is a common factor that can be eliminated.

Ex. 3: Simplify

348

11

41

4

xx

xx

4

3

x

x ← Simplify

Model Problems5

31)1

2

x

x

12)

1y

yy

2433)

363

x

x

4)1 1

x yx

x y

2 65)

2 3

k k

k k

71

26)

31

2

y

y

Homework

• Practice Sheet