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Section 7.1
Multiplication and Division with
Rational Expressions
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Objectives
o Determine any restrictions on the variable in a
rational expression.
o Reduce rational expressions to lowest terms.
o Multiply rational expressions.
o Divide rational expressions.
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Introduction to Rational Expressions
Rational Expressions
A rational expression is an algebraic expression that
can be written in the form
P
where P and Q are polynomials and Q  0.
Q
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Introduction to Rational Expressions
Notes
Remember, the denominator of a rational expression
can never be 0. Division by 0 is undefined.
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Example 1: Finding Restrictions on the Variable
Determine what values of the variable, if any, will make
the rational expression undefined. (These values are
called restrictions on the variable.)
5
a.
3x  1
Solution 3x  1  0
3x  1
1
x
3
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Set the denominator equal to 0.
Solve the equation.
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Example 1: Finding Restrictions on the Variable
(cont.)
1
5
Thus the expression
is undefined for x  . Any
3
3x  1
other real number may be substituted for x in the
1
expression. We write x  to indicate the restriction
3
on the variable.
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Example 1: Finding Restrictions on the Variable
(cont.)
x 4
b. 2
x  5x  6
2
Solution
x 2  5x  6  0
Set the denominator equal to 0.
 x  6 x  1  0
Solve the equation by factoring.
x  6  0 or x  1  0
x  1
x6
Thus there are two restrictions on the variable:
6 and –1. We write x ≠ 1, 6.
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Example 1: Finding Restrictions on the Variable
(cont.)
x 3
c. 2
x  36
Solution x 2  36  0
x 2  36
Set the denominator equal to 0.
Solve the equation.
However there is no real number whose square is 36.
Thus there are no restrictions on the variable.
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Introduction to Rational Expressions
Notes
Comments about the Numerator Being 0
If the numerator of a rational expression has a value
of 0 and the denominator is not 0 for that value of the
variable, then the expression is defined and has a
value of 0. If both numerator and denominator are 0,
then the expression is undefined just as in the case
where only the denominator is 0.
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Introduction to Rational Expressions
Summary of Arithmetic Rules for Rational Numbers
(or Fractions)
A fraction (or rational number) is a number that can
a
be written in the form where a and b are integers
b
and b  0. (Remember, no denominator can be 0.)
a a k
The Fundamental Principle: 
where b , k  0
b b k
a b
a b
The reciprocal of is and   1 where a , b  0
b a
b a
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Introduction to Rational Expressions
Summary of Arithmetic Rules for Rational Numbers
(or Fractions) (cont.)
a c ac
where b , d  0
Multiplication:  
b d b d
a c a d
Division:
   where b , c , d  0
b d b c
a c a+c
 
where b  0
Addition:
b b
b
a c a c
 
where b  0
Subtraction:
b b
b
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Introduction to Rational Expressions
The Fundamental Principal of Rational Expressions
P
If is a rational expression and P, Q, and K are
Q
polynomials where Q, K ≠ 0, then
P P K

.
Q QK
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Example 2: Reducing Rational Expressions
Use the fundamental principle to reduce each
expression to lowest terms. State any restrictions on
the variable by using the fact that no denominator can
be 0. This restriction applies to denominators before
and after a rational expression is reduced.
2 x  10
a.
3x  15
2 x  10 2  x  5 2

Solution
  x  5
3x  15 3 x  5 3 Note that x  5 is a common factor.
The key word here is factor. We
reduce using factors only.
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Example 2: Reducing Rational Expressions
(cont.)
x 3  64
b. 2
x  16
Solution
x  64

2
x  16
3

2
x

4
x

  4 x  16
 x  4 x  4

x 2  4 x  16 x  4,4



x4
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Reduce. The common factor is x – 4.
Note that x 3  64 is the difference
of two cubes. Also, note that
x 2  4 x  16 is not factorable.
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Example 2: Reducing Rational Expressions
(cont.)
y  10
c.
10  y
y  10 y  10

Solution
10  y  y  10
1 y  10

1 y  10
Note that the expression 10  y
is the opposite of y  10. When
nonzero opposites are divided,
the quotient is always 1.
1

 1  y  10
1
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Reducing (or Simplifying) Rational Expressions
Opposites in Rational Expressions
P
For a polynomial P,
 1 where P  0.
P
a  x   x  a
In particular,

 1 where x  a.
x a
x a
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Reducing (or Simplifying) Rational Expressions
Notes
COMMON ERROR
“Divide out” only common factors.
INCORRECT
4x  8
8
8 is not a common factor.
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INCORRECT
x
2
3
x 9
x3
3 and are not common
factors.
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Reducing (or Simplifying) Rational Expressions
Notes (cont.)
CORRECT
CORRECT
4 x  8 4  x  2

8
8
2
4 is a common factor
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x 2  9  x  3  x  3

x 3
 x  3
x  3 is a common factor
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Multiplication with Rational Expressions
To multiply any two (or more) rational expressions,
1. Completely factor each numerator and
denominator.
2. Multiply the numerators and multiply the
denominators, keeping the expressions in factored
form.
3. “Divide out” any common factors from the
numerators and denominators. Remember that no
denominator can have a value of 0.
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Multiplication with Rational Expressions
Multiplication with Rational Expressions
If P, Q, R, and S are polynomials and Q, S ≠ 0, then
P R P R
 
.
Q S QS
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Example 3: Multiplication with Rational
Expressions
Multiply and reduce, if possible. Use the rules for
exponents when they apply. State any restrictions on
the variable(s).
5
3
2
3 2
52 3 x  y
5x y 6 x y

a.

2
7
3
4
3

3

3

5

x

y
9 xy 15xy
3
2 x 5 2 y 3  7
2 x 3y 4 2 x x  0, y  0

 4 


9y
9
9
x x 2  4 x  x  2 x  2
x 2
 x  0,2 
b.
 2 

2
x 2 x
x
x

2
x
 
x
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Example 3: Multiplication with Rational
Expressions (cont.)
 x 1
3x  3 x  2 x  1
3 x  1 x  1
c. 2
 2

x  x 3x  6 x  3 x  x  1  3 x  12
2
2
 x 1
x 1
x 1

or 2
x  x  1
x x
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 x  1,0,1
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Example 3: Multiplication with Rational
Expressions (cont.)
x 2  7 x  12
x2  4
d.
 2
2x  6
x  2x  8
x  4  x  3 x  2 x  2


2  x  3 x  4  x  2
x  3 x  2
x 2  5x  6
x 2  5x  6


or
or
2  x  3
2  x  3
2x  6
 x  3, 2,4
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Multiplication with Rational Expressions
Notes
As shown in Examples 3c and 3d there may be more
than one correct form for an answer. After a rational
expression has been reduced, the numerator and
denominator may be multiplied out or left in factored
form. Generally, the denominator will be left in
factored form and the numerator multiplied out. As
we will see in the next section, this form makes the
results easier to add and subtract. However, be aware
that this form is just an option, and multiplying out the
denominator is not an error.
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Division with Rational Expressions
Division with Rational Expressions
If P, Q, R, and S are polynomials and Q, R, S ≠ 0, then
P R P S
   .
Q S Q R
S
R
Note that is the reciprocal of .
R
S
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Example 4: Division with Rational Expressions
12 x 2y 3x 4 y
a.
 3
2
10 xy
xy
2
4
2
3
12
x
y
3
x
y
12
x
y
xy
Solution
 3 
 4
2
2
10 xy
xy
10 xy 3x y
2  2  3 x3  y4

5
3
2 5 3 x  y
2 x 3 5 y 4 3
Note that in this example

we have used the quotient
5
rule for exponents.
2
2 x y 2y
 2

5x
5
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Example 4: Division with Rational Expressions
(cont.)
x3  y3 y  x
b.

3
x
xy
Solution
3
3
3
3
x  y y  x x  y xy



3
3
x
xy
x
yx

1

x


2
2
x

y
x

xy

y
xy
 
2
x3 y  x

 y x 2  xy  y 2
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x
2
Note that
   x y  xy
2
x
2
xy
 1.
yx
 y3
2
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Example 4: Division with Rational Expressions
(cont.)
x 2  8 x  15 2 x 2  5x  3
c.

2
2 x  11x  5
4 x2  1
Solution
x 2  8 x  15 2 x 2  5x  3
x 2  8 x  15
4 x2  1

 2
 2
2
2
2 x  11x  5
4x  1
2 x  11x  5 2 x  5x  3
x  3 x  5 2 x  1 2 x  1


2x  1 x  5 x  32x  1
Remember that you have the
x  5 2 x  1 2 x 2  11x  5 option of leaving the numerator



and/or denominator in factored
2x  1 x  5 2x  1 x  5 form.
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Practice Problems
Reduce to lowest terms. State any restrictions on the
variables.
1. 5x  20
7 x  28
2. 6  3x
3x  6
3. x 2  x  2
2
x  3x  2
Perform the following operations and simplify the
results. Assume that no denominator has a value of 0.
x 7
x2
y2  y  6
y2  4
4.
5. 2

 2
3
2
x
49  x
y  5y  6 y  4 y  4
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Practice Problems (cont.)
x 2  3x x 2  3
6.

2x  1
x 1
x 2  2x  3 2x 2  9 x  5 4 x  2
7. 2
 2
 2
x  3x  10 x  2 x  1 x  x
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Practice Problem Answers
1. 5 , x  4
7
1
4.
x  x  7
2. 1, x  2
3. x  1 , x  2, 1
x 1
5. 1
2
x
x
6.
2x  1
2
x

3
x
7.
2  x  2
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