9.1 Multiplying and Dividing Rational Expressions

©2001 by R. Villar

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Multiplying and Dividing Rational Expressions

A rational expression is considered to be simplified when its numerator and denominator have no common factors left…

Example: Simplify8x 3 2x 2

4x 2 xFactor the numerator and denominator...

2x2 (4x 1)

x(4x 1)

Cancel common factors...

2x2

xReduce... 2x

Multiplying rational expressions works the same as multiplying fractions…you may multiply numerators and denominators and then simplify, or you can cross cancel common factors...

Example: Multiply 8x3

3xy•

6y4

12x 2

With monomials, it’s a good idea to multiply first, and then simplify...

48x3y4

36x3yNow reduce...

4y3

3

48

36•

x3

x3 •y4

y

4

3

With polynomials, factor each numerator and denominator, if factorable, and cross cancel to simplify...

Example: Multiply 2x2 x 3

x 2 x•

x 1

2x 3Factor...

(2x 3)(x 1)

x(x 1)•

(x 1)

(2x 3)Cross cancel...

x + 1 x

Recall that to divide fractions, you can “invert and multiply: ...

Example: Divide x3 3x2

3x 6

x3 8x 2 15x

6x 2 18x 60Rewrite as multiplication...

x2 (x 3)

3(x 2)•

6(x 2 3x 10)

x(x2 8x 15)

Cross cancel...

= 6x2

3x

x3 3x2

3x 6•

6x2 18x 60

x3 8x2 15xFactor...

x2 (x 3)

3(x 2)•

6(x 5)(x 2)

x(x 5)(x 3)

= 2x