1

Topic 8.2.1Topic 8.2.1

Multiplying RationalExpressions

Multiplying RationalExpressions

2

Lesson

1.1.1

California Standard:13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.

What it means for you:You’ll multiply rational expressions by factoring and cancelling.

Topic

8.2.1

Key words:• rational• nonzero• common factor

Multiplying Rational ExpressionsMultiplying Rational Expressions

3

Lesson

1.1.1

In Section 8.1 you learned about simplifying rational expressions.

Topic

8.2.1

In this Topic you’ll learn to multiply rational expressions, but then you’ll use the same simplification methods to express your solutions in their simplest forms.

Multiplying Rational ExpressionsMultiplying Rational Expressions

4

Lesson

1.1.1

Multiplying by Rational Expressions

Topic

8.2.1

Given any nonzero expressions m, c, b, and v:

That is, the product of two rational expressions is the product of the numerators divided by the product of the denominators.

Multiplying Rational ExpressionsMultiplying Rational Expressions

5

Lesson

1.1.1

Multiplying by Rational Expressions

Topic

8.2.1

These expressions can often be quite complicated, so simplify them as much as you can before multiplying.

• First, factor the numerators and denominators (if possible).

• Then find any factors that are common to both the top line (the numerator) and the bottom line (the denominator) and cancel them before multiplying.

Multiplying Rational ExpressionsMultiplying Rational Expressions

6

Example 1

Topic

8.2.1

Simplify .Solution

Step 1: Factor the numerators and denominators.

Solution follows…

Step 2: Cancel all the common factors and multiply.

=1

1

1

11

1

=

=

Multiplying Rational ExpressionsMultiplying Rational Expressions

7

Example 2

Topic

8.2.1

Solution

Step 1: Factor the numerators and denominators if possible.

Solution follows…

= –

Step 2: Cancel all the common factors and multiply.1

112

Multiply and simplify .

=

=

=1

1

1

1

1

1

Multiplying Rational ExpressionsMultiplying Rational Expressions

8

Multiply and simplify the rational expressions.

1. 2.

3. 4.

5. 6.

Lesson

1.1.1

Guided Practice

Topic

8.2.1

Solution follows…

Multiplying Rational ExpressionsMultiplying Rational Expressions

9

Lesson

1.1.1Topic

8.2.1

You can extend this concept to the multiplication of any number of rational expressions, with any number of variables.

Multiplying Rational ExpressionsMultiplying Rational Expressions

10

Example 3

Topic

8.2.1

Solution

Factor the numerators and denominators, and cancel all the common factors and multiply.

Solution follows…

= or

Simplify .

=

=

1 1 1 1 1

1 1 1 1 1

1

1

Multiplying Rational ExpressionsMultiplying Rational Expressions

11

Example 4

Topic

8.2.1

Solution

Factor the numerators and denominators, then cancel all the common factors and multiply.

Solution follows…

Simplify .

= or

= 1

1

1 1 1

1 1 1

Multiplying Rational ExpressionsMultiplying Rational Expressions

12

Multiply and simplify the rational expressions.

7.

8.

9.

Lesson

1.1.1

Guided Practice

Topic

8.2.1

Solution follows…

x – 5

2(m + 1) or 2m + 2

Multiplying Rational ExpressionsMultiplying Rational Expressions

13

Multiply and simplify the rational expressions.

1.

2.

3.

4.

Independent Practice

Solution follows…

Topic

8.2.1

1

Multiplying Rational ExpressionsMultiplying Rational Expressions

14

Multiply and simplify the rational expressions.

5.

6.

7.

Independent Practice

Solution follows…

Topic

8.2.1 Multiplying Rational ExpressionsMultiplying Rational Expressions

15

Topic

8.2.1

Round UpRound Up

Multiplying Rational ExpressionsMultiplying Rational Expressions

Usually the most difficult thing when solving problems like these is factoring the numerators and denominators.

Look for “difference of two squares” expressions, perfect squares, and minus signs that you can factor outside the parentheses.