Multiplying and Dividing Rational Expressions

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Multiplying and Dividing Rational Expressions
8-2
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Simplify each expression. Assume all
variables are nonzero.
1. x5 ? x2
x7
2. y3 ? y3
y6
3.
4.
x4
Factor each expression.
5. x2 2x 8
(x 4)(x 2)
6. x2 5x
x(x 5)
7. x5 9x3
x3(x 3)(x 3)
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Objectives
Simplify rational expressions. Multiply and
divide rational expressions.
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Vocabulary
rational expression
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(No Transcript)
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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.
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Example 1A Simplifying Rational Expressions
Simplify. Identify any x-values for which the
expression is undefined.
Quotient of Powers Property
The expression is undefined at x 0 because this
value of x makes 6x4 equal 0.
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Example 1B Simplifying Rational Expressions
Simplify. Identify any x-values for which the
expression is undefined.
Factor then divide out common factors.

The expression is undefined at x 1 and x 3
because these values of x make the factors (x
1) and (x 3) equal 0.
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Example 1B Continued
Check Substitute x 1 and x 3 into the
original expression.


Both values of x result in division by 0, which
is undefined.
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Check It Out! Example 1a
Simplify. Identify any x-values for which the
expression is undefined.
Quotient of Powers Property
The expression is undefined at x 0 because this
value of x makes 8x2 equal 0.
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Check It Out! Example 1b
Simplify. Identify any x-values for which the
expression is undefined.
Factor then divide out common factors.

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Check It Out! Example 1b Continued

Both values of x result in division by 0, which
is undefined.
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Check It Out! Example 1c
Simplify. Identify any x-values for which the
expression is undefined.
Factor then divide out common factors.

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Check It Out! Example 1c Continued
Both values of x result in division by 0, which
is undefined.
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Example 2 Simplifying by Factoring by 1
Simplify . Identify any x
values for which the expression is undefined.
Factor out 1 in the numerator so that x2 is
positive, and reorder the terms.
Factor the numerator and denominator. Divide out
common factors.
Simplify.
The expression is undefined at x 2 and x 4.
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Example 2 Continued
Check The calculator screens suggest that

except when x 2 or x 4.
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Check It Out! Example 2a
Simplify . Identify any x
values for which the expression is undefined.
Factor out 1 in the numerator so that x is
positive, and reorder the terms.
Factor the numerator and denominator. Divide out
common factors.
Simplify.
The expression is undefined at x 5.
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Check It Out! Example 2a Continued
Check The calculator screens suggest that
2 except when
x 5.
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Check It Out! Example 2b
Simplify . Identify any x
values for which the expression is undefined.
Factor out 1 in the numerator so that x is
positive, and reorder the terms.
Factor the numerator and denominator. Divide out
common factors.
Simplify.
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Check It Out! Example 2b Continued
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You can multiply rational expressions the same
way that you multiply fractions.
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Example 3 Multiplying Rational Expressions
Multiply. Assume that all expressions are defined.
A.
B.
3
5
3
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Check It Out! Example 3
Multiply. Assume that all expressions are defined.
A.
B.
2
2
2
3
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You can also divide rational expressions. Recall
that to divide by a fraction, you multiply by its
reciprocal.
2

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Example 4A Dividing Rational Expressions
Divide. Assume that all expressions are defined.
Rewrite as multiplication by the reciprocal.
2
3
3
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Example 4B Dividing Rational Expressions
Divide. Assume that all expressions are defined.
Rewrite as multiplication by the reciprocal.
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Check It Out! Example 4a
Divide. Assume that all expressions are defined.
Rewrite as multiplication by the reciprocal.
3
1
x4 y
2
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Check It Out! Example 4b
Divide. Assume that all expressions are defined.
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Example 5A Solving Simple Rational Equations
Solve. Check your solution.
Note that x ? 5.
x 5 14
x 9
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Example 5A Continued
Check
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14 14
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Example 5B Solving Simple Rational Equations
Solve. Check your solution.
Note that x ? 2.
x 5 7
x 2
Because the left side of the original equation is
undefined when x 2, there is no solution.
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Example 5B Continued
Check A graphing calculator shows that 2 is
not a solution.
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Check It Out! Example 5a
Solve. Check your solution.
Note that x ? 4.
x 3 7
x 4
Because the left side of the original equation is
undefined when x 4, there is no solution.
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Check It Out! Example 5a Continued
Check A graphing calculator shows that 4 is
not a solution.
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Check It Out! Example 5b
Solve. Check your solution.
2x 3 5
x 4
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Check It Out! Example 5b Continued
Check
5
5 5
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Lesson Quiz Part I
Simplify. Identify any x-values for which the
expression is undefined.
1.
x ? 2, 5
x ? 1, 6
2.
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Lesson Quiz Part II
Multiply or divide. Assume that all expressions
are defined.
3.
4.
Solve. Check your solution.
5.
x 4