1.5.b rational expressions

Chapter 1.5

Rational Expressions

1


are algebraic

expressions that are quotients of

po

Rational Exp

lynomials

res ns

,

io

.

s

0P x

Q xQ x

2


A rational expression is in its

if the polynomials

in the numerator and denominator

are relatively pri

simplest fo

me.

, where , 0

rm

ac ab c

bc b

3

Example 1.5.1

2

Express the following in simplest

form.

2 241.

2 2

2, 2

x xx

x x

x x

4

Example 1.5.1

2

2

Express the following in simplest form.

5 712 352.

3 10 5 2

7, 5

2

x xx x

x x x x

xx

x

5

Example 1.5.1

23

2

3 3 9273.

3 3

3 9, 3

x x xx

x x

x x x

6

Similar Rational Expressions

2

2 2 3 3

are

those that have the same denominat

Similar

or.

3

rational expressi

1 1

ons

1

x x x

x x x x

Similar NOT Similar 7

Example 1.5.2

2

2

Write the following as similar

rational expressions.

3 21. , : 1 1

1 1

3 13 3 3

1 1 1 1

2 12 2 2

1 1 1 1

LCD x xx x

x x

x x x x

x x

x x x x

8

Example 1.5.2

3

3

2

3

3

52. , : 2

2 2

2

2 2

5

2

xLCD x

x x

x xx

x x

x

9

Example 1.5.2

2 13. , : 3

3 3

1 11 1 1

3 3 3 1 3

2

3

LCD xx x

x x x x

x

10

Addition of Rational

Expressions

Definition

Let , , and be polynomials

in then

0

P x Q x R x

x

P x R x P x R x

Q x Q x Q x

Q x

11

Example 1.5.3

2

2

Perform the indicated operation.

11. : 2 2

2 4

1

2 2 2

2

2 2 2 2

2 2 2 2

2 2 4

xLCD x x

x x

x

x x x

x x

x x x x

x x

x x x 12

Example 1.5.3

3

3

2

3

2

3

2 2

3 2 2

12. : 1

1 1

1 1

1

2 1

1

1 1

3 3 11

xLCD x

x x

x x

x

x x x

x

x x x x

x x xx13

Multiplication of Rational

Expressions

Definition

Let , , , and be

polynomials in then

, 0

P x Q x R x S x

x

P x R x P x R x

Q x S x Q x S x

Q x S x

14

Example 1.5.4

2

Perform the indicated operation.

3 21.

1 1

6

1 1

6

1

x x

x x

x

15

Example 1.5.4

2 62.

2 3

3 2

2 3

, 3, 2

x x x

x x

x xx

x x

x x

16

Example 1.5.4

2 2

3

2

3

2 1 13.

1 1

1 1 1

1 1

1, 1

1

x x x

x x

x x x

x x

xx

x

17

Example 1.5.4

2

2

2 4 24.

2 4 8

2 4 2

2 4 2

2 2 2 2

2 4 2

2, 2

2

x x

x x

x x

x x

x x x

x x

xx

18

Subtraction and Division

P x R x P x R x

Q x S x Q x S x

P x

Q x P x S x

R x Q x R x

S x

19

Example 1.5.5

2 2

Perform the indicated operations.

3 41. : 2 1 2

2 1 2

3 2 4 2 1

2 1 2

5 6 2 7 4

2 1 2

x xLCD x x

x x

x x x x

x x

x x x x

x x

20

Example 1.5.5

2 2

2 2

2 2

2

5 6 2 7 4

2 1 2

5 6 2 7 4

2 1 2

2 10 2 10

2 1 2 2 3 2

x x x x

x x

x x x x

x x

x x x x

x x x x

21

Example 1.5.5

2

2

2

2

1 12. : 3 3

3 3 9

1 1

3 3 3 3

1 3 3 1

3 3

3 3 1

3 3

4 4

9

xLCD x x

x x x

x

x x x x

x x x

x x

x x x

x x

x x

x

22

Example 1.5.5

2 2

2 2

2 2

2 2

2

2

2

2 4 43.

6 9

2 9

6 4 4

2 1 3 3

3 2 2

1 3, 2,3

2 2

4 4

2 2

2 2, 2

2 2 2

x x x x

x x x

x x x

x x x x

x x x x

x x x

x xx

x x

x x

x x

x xx

x x x

23

Example 1.5.5

22 2

2

2 2

1 2 22

4.3 1 3 1 3 1

2, 0

3 1

xx xx x x

x x x

x x x

xx

x

24

Engineering

1.5.b rational expressions