§ 6.3

Complex Rational Expressions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.3

Simplifying Complex Fractions

Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more fractions.

.1

51

55

x

xx

Woe is me,

for I am complex.

I am currently suffering from…a feeling of complexity

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.3

Complex Rational Expressions

Simplifying a Complex Rational Expression by Multiplying by One in

the Form T

1) Find the LCD of all rational expressions within the complex rational expression.

2) Multiply both the numerator and the denominator of the complex rational expression by this LCD.

3) Use the distributive property and multiply each term in the numerator and denominator by this LCD. Simplify. No fractional expressions should remain.

4) If possible, factor and simplify.

LCD

LCD

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.3

Simplifying Complex Fractions

EXAMPLEEXAMPLE

Simplify: .1

51

55

x

xx

SOLUTIONSOLUTION

The denominators in the complex rational expression are 5 and x. The LCD is 5x. Multiply both the numerator and the denominator of the complex rational expression by 5x.

x

xx

x

x

x

xx

151

55

5

51

51

55 Multiply the numerator and

denominator by 5x.

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.3

Simplifying Complex Fractions

xxx

xx

xx

15

51

5

55

55

Use the distributive

property.

CONTINUECONTINUEDD

xxx

xx

xx

15

51

5

55

55

Divide out common factors.

5

252

x

xSimplify.

51

55

x

xxFactor and simplify.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.3

Simplifying Complex Fractions

Simplify.

CONTINUECONTINUEDD

1

5x

Simplify.5x

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.3

Simplifying Complex Fractions

EXAMPLEEXAMPLE

Simplify: .6

16

1xx

SOLUTIONSOLUTION

The denominators in the complex rational expression are x + 6 and x. The LCD is (x + 6)x. Multiply both the numerator and the denominator of the complex rational expression by (x + 6)x.

Multiply the numerator and denominator by (x + 6)x.

6

16

1

6

6

6

16

1

xxxx

xxxx

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.3

Simplifying Complex Fractions

Use the distributive property.

66

16

61

6

xxx

xxx

xx

CONTINUECONTINUEDD

Divide out common factors.

66

16

61

6

xxx

xxx

xx

Simplify.

66

6

xx

xx

Simplify.xx

xx

366

62

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.3

Simplifying Complex Fractions

CONTINUECONTINUEDD

Subtract.xx 366

62

Factor and simplify.

66

16

xx

Simplify. 6

1

xx

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.3

Simplifying Complex Fractions

Simplifying a Complex Rational Expression by Dividing

1) If necessary, add or subtract to get a single rational expression in the numerator.

2) If necessary, add or subtract to get a single rational expression in the denominator.

3) Perform the division indicated by the main fraction bar: Invert the denominator of the complex rational expression and multiply.

4) If possible, simplify.

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.3

Simplifying Complex Fractions

EXAMPLEEXAMPLE

Simplify: .

6653

442

9

22

22

mmm

mm

mmmm

SOLUTIONSOLUTION

1) Subtract to get a single rational expression in the numerator.

222 2

2

3344

2

9

mmm

m

mmm

m

2

2

22

2

233

3322

332

332

233

2

mmm

mmmm

mmm

mm

mmm

mm

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.3

Simplifying Complex Fractions

2) Add to get a single rational expression in the denominator.

2

23

2

223

233

1846

233

18244

mmm

mmm

mmm

mmmm

3223

3

665

322

mm

m

mmmm

m

mm

323

333

332

3

323

33

mmm

mmm

mmm

mm

mmm

m

323

9

323

393 22

mmm

m

mmm

mmm

CONTINUECONTINUEDD

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.3

Simplifying Complex Fractions

3) & 4) Perform the division indicated by the main fraction bar: Invert and multiply. If possible, simplify.

3239

233

1846

6653

442

92

2

23

22

22

mmmm

mmm

mmm

mmm

mm

mmmm

9

323

233

184622

23

m

mmm

mmm

mmm

9

323

233

184622

23

m

mmm

mmm

mmm

92

18462

23

mm

mmm

CONTINUECONTINUEDD

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.3

Simplifying Complex Fractions

Important to Remember:

Complex rational expressions have numerators or denominators containing one or more fractions.

Complex rational expressions can be simplified by one of two methods presented in this section:

(a) multiplying the numerator and denominator by the LCD(b) obtaining single expressions in the numerator and denominator and then dividing, using the definition of

division for fractions