12.1 – Simplifying Rational Expressions

A rational expression is a quotient of polynomials.

For any value or values of the variable that make the denominator zero, the rational expression is considered to be undefined at those value(s).

3

5 1

x

x

23 6 7

2 6

x x

x

4

3

4 4 12

q

q q

The denominator can not equal zero.

What are the values of the variable that make the denominator zero and the expression undefined?

3

5 1

x

x

23 6 7

2 6

x x

x

2

3 2

12

x

x x

5 1 0x

5 1x 1

5x

2 6 0x

2 6x 3x

3 4 0x x

2 12 0x x

3 0x 4 0x 3x 4x

12.1 – Simplifying Rational Expressions

Simplifying4 3

5 5

x x

x

2

5

25

q

q

3x

3

5

x

5q

1

5q

1x

5 1x 5q 5q

12.1 – Simplifying Rational Expressions

Simplifying2

2

11 18

2

x x

x x

2x

9

1

x

x

9x

2x 1x

12.1 – Simplifying Rational Expressions

Simplifying4

4

x

x

4x

4x

4x

1 4x 1

12.1 – Simplifying Rational Expressions

12.2 - Rational Expressions –Multiplication and Division

3 2

3

5 2

3 15

a b

b a

10

2

3a 2b45 a 3b

2a9 b

Multiplication:

Multiplication: 2

3 2

3 6 7

14 2

x x

x x

3

21

2x 2x

27x14 2x

1432

12.2 - Rational Expressions –Multiplication and Division

19

253

147

842

2

2

x

xx

xx

x

4 2x

7x 2x 3 1x

3 1x 3 1x 2x

4 2x

7x 3 1x

Multiplication:

12.2 - Rational Expressions –Multiplication and Division

y

xx

26

7 2

27

6

x 1

3

7xy

2y

x

1

3

Division:

12.2 - Rational Expressions –Multiplication and Division

2

123

6

4 2

xx

24

6

x

24

6

x

4

3

x 9

4x

2

3 12x

23 4x

1

1

3

1

3

Division:

12.2 - Rational Expressions –Multiplication and Division

2

25

4

410 23

2

x

xx

x

x

2

2

10 4

4

x

x

2

2x 2

22 xx

3 2

2

5 2

x

x x

5 2x

2x 2x 2x

2x 5 2x

2

1

x

Division:

12.2 - Rational Expressions –Multiplication and Division

21

129

147

8103 2

x

x

xx

21

21

23 10 8

7 14

x x

x

2x

1

7

21

9 12x

3 4x

7 2x 21

3 3 4x

21

31

Division:

12.2 - Rational Expressions –Multiplication and Division