Transcript Lesson 6.2A
Multiplying and Dividing Rational Expressions
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How do we simplify rational expressions?
How do we multiply and divide rational expressions?
Multiplying and Dividing Rational Expressions
In Lesson 8-1, you worked with inverse variation functions such as y = . The expression on the right side of this equation is a rational expression. A rational expression is a quotient of two polynomials. Other examples of rational expressions include the following:
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions
Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator.
Caution!
When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 1: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
5
10x 8
3
6x 4
x
4 5
x
4 3 The expression is undefined at x = 0 because this value of x makes 6x 4 equal 0.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 2: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
Factor; then divide out
x x
x
2 + x – 2
x
2
2
+ 2x – 3
3
x x
1 1
x
x
2
common factors.
The expression is undefined at x = 1 and x = –3 because these values of x make the factors of the denominator (x – 1) and (x + 3) equal 0.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 3: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
2
16x 11
1
8x 2
x
9 2
x
9 1 2
x
9 The expression is undefined at x = 0 because this value of x makes 8x 2 equal 0.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 4: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
Factor; then divide out
3x + 4
3
x
3x 2
4
+ x – 4
x
1
x
1
common factors.
12 1 3 , 4 3 1 3 The expression is undefined at x = 1 and x = – 4 because these values of x make the factors of the denominator (x – 1) and (3x + 4) equal 0.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 5: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
Factor; then divide out
2
x
1
6x 2
+ 7x + 2
2
6x 2
x
– 5x – 6
3 3
x
2 2
x
2
x
3 1
common factors.
12 7 36 3 , 4 9 , 4 6 6 6 6 5 1 2 3 2 2 3 2 3 The expression is undefined at x =– 2 3 and x = 3 because these values of x make the factors of the denominator (3x + 2) and (2x – 3) equal 0.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 6: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
Factor; then divide out
x
2
x
x
x
2 4
4x – x 2
x
4
– 2x – 8
x
x
x common factors.
The expression is undefined at x = –2 and x = 4.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 7: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
Factor; then divide out
2 2
x x
5 1 0
10 – 2x x – 5
2 2
common factors.
The expression is undefined at x = 5.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 8: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined.
Factor; then divide out
x
x
– x 2 + 3x
2
2x 2
x
– 7x + 3
1 2
x
1
common factors.
6 1 , 2 6 2 3 7
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions
You can multiply rational expressions the same way that you multiply fractions.
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 9: Multiplying Rational Expressions Multiply. Assume that all expressions are defined.
3x 5
y
3 2x 3
y
7
5 30
x x
3
y
10x 3
y
4 9x 2
y
5
3 18
x
5
y
12
y
5 5
x
3 3
y
5
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions Example 10: Multiplying Rational Expressions Multiply. Assume that all expressions are defined.
x
15
x
7 2x
2 20
x x
8 3
20
x
4
3 30
x
5 2
x
3 3
Holt McDougal Algebra 2
Multiplying and Dividing Rational Expressions
Lesson 6.2 Practice A
Holt McDougal Algebra 2