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Multiplying and Dividing
Expressions
Section 8-2
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10.3 Multiplying and Dividing Expressions
1
Quick Review
• Factor x  6 x  9
2
( x  3)( x  3)
• Factor x 2  9
( x  3)( x  3)
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10.3 Multiplying and Dividing Expressions
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Rules of Multiplying Expressions
1. Take GCF of numerator(s) and denominator(s) if
possible.
2. Factor the numerator(s) (top) into simplest form
3. Factor the denominator(s) (bottom) into the simplest
form
4. Cancel [bottom or cross] out the factors if needed,
multiply and simplify. Apply the GCF to the factors,
then…
FACTOR,
CANCEL,
QUIT.
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10.3 Multiplying and Dividing Expressions
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Example 1
Simplify
x2  6 x  9
x2  9
Step 1: Is there a GCF?
Step 2: Factor the numerator
(s) of x2 + 6x + 9
Step 3: Factor the
denominator(s) x2 - 9
Step 4: Cancel out the factors
if needed and then multiply.
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10.3 Multiplying and Dividing Expressions
No
(x + 3) (x + 3)
(x + 3) (x - 3)
( x  3)( x  3) ( x  3)

( x  3)( x  3) ( x  3)
4
Example 2
Simplify
x 2  3x  2
x2  5x  6
( x  1)( x  2) ( x  1)

( x  2)( x  3) ( x  3)
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10.3 Multiplying and Dividing Expressions
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Example 3
Simplify
3x 2 2 y 2

4 y 6x
xy
4
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Example 4
Simplify
4 x2  4 x x2  x  6

2
x  2x  3
4x
( x  2)
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10.3 Multiplying and Dividing Expressions
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Undefined Values
0

1
0
1
 undefined
0
Any substituted number into a variable which the
expression in the denominator equal to zero, it will
cause the equation to be undefined.
Quick Steps:
–
–
–
After factoring an expression, put all factors equal to zero
Solve for the variable
Answers are the restrictions
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10.3 Multiplying and Dividing Expressions
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Example 5
3x  4
Simplify 2
. Identify any x-values for which the
3x  x  4
expression is undefined.
(3x + 4)
1
=
(3x + 4)(x – 1)
(x – 1)
3x  4  0
4
x
3
x 1  0
x 1
1
4
, x   ,1
( x  1)
3
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Example 6
6 x2  7 x  2
.
2
6 x  5x  6
Simplify
Identify any x-values for which
the expression is undefined.
(2x + 1)
(2x – 3)
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The expression is undefined at x = –2/3 and x = 3/2 because these
values of x make the factors (3x + 2) and (2x – 3) equal 0.
10.3 Multiplying and Dividing Expressions
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Example 7
4 x  x2
x2  2 x  8
Simplify
. Identify any x-values for which the
expression is undefined.
–1(x2 – 4x)
x2 – 2x – 8
Factor out –1 in the numerator so that
x2 is positive, and reorder the terms.
–1(x)(x – 4)
(x – 4)(x + 2)
Factor the numerator and denominator.
Divide out common factors.
–x
(x + 2 )
Simplify.
The expression is undefined at x = –2 and x = 4.
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10.3 Multiplying and Dividing Expressions
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Rules of Divide Expressions
1.
2.
3.
4.
5.
Take GCF of numerator(s) and
denominator(s) if possible.
Take the reciprocal (flip) of the second
fraction. -- KEEP IT, CHANGE IT, FLIP IT -Factor the numerator(s) (top) into simplest
form
Factor the denominator(s) (bottom) into
the simplest form
Cancel out the factors and cancel if needed,
multiply and simplify.
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Example 8
Simplify
3x  12 6 x  24

2x  4 4x  8
3( x  4) 6( x  4)
Step 1: Is there a GCF?
2( x  2) 4( x  2)
Step 2: Take the reciprocal (flip) of the
second fraction.
Step 3: Factor the numerator(s) (top) into
simplest form
Step 4: Factor the denominator(s) (bottom)
into the simplest form
Step 5: Cancel out the factors if needed,
multiply and simplify.
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10.3 Multiplying and Dividing Expressions
6  x  4
4  x  2

4  x  2
6  x  4
3 x  4
2  x  2
4  x  2
6  x  4
3 x  4 4  x  2

1
2  x  2 6  x  4
13
Example 9
x

Simplify
2
 1
 x  1
 ( x  1)
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Example 10
Simplify
x 2  8 x  16 x 2  6 x  8

x2
x2  4
( x  4)( x  2)

( x  2)
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Example 11
Simplify
( x  4)
5 x  20

3x 2  4 x  4 ( x  2)(7 y  5)
(7 y  5)

5(3x  2)
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Example 12
Solve. Check your solution and list restrictions of
(x + 5)(x – 5) = 14
(x – 5)
x + 5 = 14
x 2  25
 14
x 5
Note that x ≠ 5.
x=9
x  9; x  5
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Example 13
Solve. Check your solution and list restrictions of
x 2  3x  10
7
x2
No Solution; x  2
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Your Turn
Simplify and/or solve these problems
1.
6x –
x2 – 7x + 6
x2
2.
x  2 2x  6

x3 x5
2
2
x

2
x

3
x
 2 x  15
4.
 2
2
x x2
x  x6
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5.
5
4
3. x y  1
3
3xy x y
4x2 – 9
=5
2x + 3
10.3 Multiplying and Dividing Expressions
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Assignment
Pg 580: 1-9 odd, 19-27 odd,
37, 39, 42
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