e - book for college algebra king fahd university of petroleum & minerals b.2 e - book for...

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

B.2


Rational Expressions

KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

Algebra of rational expressions

Reduction of rational expressions

Complex rational expressions


Definition of Rational Expression


A rational expression in is a rationx

p x

R xq x

Where p( x ) and q ( x ) are polynomials in and

.

x 0q x


Note:

Since an integer is a polynomial of degree zero, every rational number is a rational expression. Other examples of rational expressions are


2 1

2 5

x xR x

x

2

3

2 5 6

8 1

a aW a

a a

2

2

5 6

1

y yQ y

y


Definition:


The domain of the rational function p x

R xq x

consists of all real numbers except those for which q (x ) = 0

Definition

The value of the rational expression p x

R xq x

when x is equal to a is obtained by replacing x by a in R ( x ).



Example 1

3 2

2 1

xR x

x

find

1

2R

Solution

3 2

2

11 2

12 12

R

32

21 1

722

7

4



Example 2

Find the domain of 2

2

3 2 1

1

x xQ x

x

Solution

The domain of Q( x ) is all real numbers except x = 1 and -1

2

22 3 2 1

( 1

2

)

1

(

3

11 )

x x

x

xx

x

xQ

x

( , 1) ( 1,1) (1, )


Reducing Rational Expressions


Reducing p x

R xq x

consists of two steps:

Step 1 Factor both and p x q x

Step 2 Divide both by the greatest common factor.

and p x q x



Example 3 Reduce to lowest terms

2

2

3 9 61.

2 8

x x

x

2

2

3 3 2

2 4

x x

x

3 2 1

2 2 2

x x

x x

3 1,

2 22

x

xx



4

3

3 32.

w w

w w

3

2

3 1

1

w w

w w

23 1 1

1 1

w w w w

w w w

23 1,

10,1, 1w

w w

w


Multiplication and Division of Rational Expressions


Multiplication:

p x r x p x r x

q x t x q x t x

Division:

p x r x p x t x p x t x

q x t x q x r x q x r x



Example 4 Perform the indicated operations and reduce the resulting rational expression.

2

2

6 8 4 4 11.

2 5 3 6 3

x x x

x x x

22 3 4 2 1

2 1 3 3 2 1

x x

x x x

2 3 4,

3

1

23x

x

x



2 2

2 2

9 6 3 32.

9 6 9

x x x xk x k

x x x

2 2

2 2

9 6 9

9 6 3 3

x x x

x x x xk x k

2

2

3 3 3

3 39

x x x

x x x k x k

3

2

3 3

3 3

x x

x x x k

2

3 ,3

,, 33

x

x x kx k


Addition and Subtraction of Rational Expressions


a c a c

b b b

Identical denominators

Different denominator

a c ad bc

b d bd

Least Common Denominator

(L.C.D)



Example 5 Find the L.C.D.

2

2and

2 15

x

x x

2

2

4 5

x

x x

Solution

21 2 15 3 5d x x x x

22 4 5 5 1d x x x x

. . . 5 1 3L C D x x x



Example 6

Solution

Perform the indicated operations and simplify.

2 2

4 3 11.

x a x

x a a x

The L.C.D.= (x – a) ( x + a)

1 3 14 x x ax a

x a x a x a x a

23 3 3x xa x

x a x a

23 x xa x

x a x a



Solution

2

4 2 22.

3 1x x x x

The L.C.D.= 3 x ( x + 1)

4 1 6 6

3 1 3 1 3 1

x x

x x x x x x

10 10

3 1

x

x x

10 1 1

3 1 3

x

x x x

Example 6. (cont.)


Complex Rational Expressions:


An algebraic expression which has rational expressions in its numerator or denominator is called a complex rational expression. For example,

1 22 3 , and 1 34 8

1 22 2

3 42 2

x x

x x


Complex Rational Expressions:


To simplify such a complex expression we multiply both its numerator and denominator by the L.C.D. of all the algebraic expressions involved in the makeup of the complex expression.

Example 7

Simplify the following complex rational expressions:

2

2 23 3)2 1

3 6

xa

x

2 2

2 2

1

)1 3

ax a x ab

a x a x



Solution

2

2 23 3)2 1

3 6

xa

x

2

2 23 32 1

3 6

66

x

x

xx

2

2

2 4

4

x x

x

2 2

2 2

x x

x x

2

2

x

x



Solution

2 2

2 2

1

)1 3

ax a x ab

a x a x

. .L C D x a x a

2 2

2 2

1

1 3

ax a x a

a x a x

x a x a

x a x a

2

3 33

a x a a x

x a x a

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