MTH 092

Section 12.1Simplifying Rational Expressions

Section 12.2Multiplying and Dividing Rational Expressions

Fractions Again?!?!?!?

• A rational expression is of the form

Where P and Q are polynomials with Q not equal to 0.

Q

P

Why Can’t Q be equal to 0?

• Recall, from our work with slope, that division by 0 is undefined.

• If Q is a polynomial, then it has variables.• Those variables cannot take on values that

cause Q to become 0.• To figure out what those values are, set Q = 0

and solve.• Key words: undefined, domain

Examples

• Find any numbers for which each rational expression is undefined:

145

111

219

25

15

3

2

2

2

3

xx

x

xx

x

x

xx

Reducing To Lowest Terms

• Recall that reducing a fraction means dividing the numerator and denominator by the same value (usually the greatest common factor).

• Reducing a rational expression involves two steps:

1.Factor both the numerator and denominator.2.Cancel common factors.• Cancel factors, not terms.

Examples: Reduce to Lowest Terms

592

144

7017

10

11

99925

5

9

9

2

2

2

34

2

2

xx

xx

xx

xx

xx

xx

x

y

y

Multiplying and Dividing

1. Factor the numerators and denominators completely.

2. Cancel common factors.3. Remember that when you are dividing, you

must multiply by the reciprocal of the second rational expression.

4. You can not cancel terms. You can not cancel parts of terms.

Multiply or Divide

32

10020

352

13512

2811

4415

209

25

155

5

3

6

5

20

244

2

2

2

2

2

2

x

x

xx

xxx

xx

xx

xx

xx

xx

x

Opposite Factors Make -1

• For all real numbers a and b,

• The -1 is usually put in the numerator and can be distributed.

1ab

ba

Apply “the rule of -1”

• Reduce, multiply, or divide as indicated:

9

12

3

327

339

7

49

9

9

2

222

5

2

x

xy

x

yx

ab

ba

x

y

y

y

y