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Simplify Example 2 1 2
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05/03/23 16:19 10.3 Multiplying and Dividing Expressions 1
Simplify,Multiply and Divide
RationalExpressions
Section 8-2
Simplify7
4
32xx
Example 1
32x x x x x x x
x x x x
33
2x
Simplify5
2
168xx
Example 2
168x x x x x
x x
32x
1
2
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 4
Example 3Simplify
2
2
6 99
x xx
Step 1: Is there a GCF?
Step 2: Factor the numerator (s) of x2 + 6x + 9Step 3: Factor the denominator(s) x2 - 9 Step 4: Cancel out the factors if needed and then multiply.
No
(x + 3) (x + 3)(x + 3) (x -
3) ( 3)( 3) ( 3)( 3)( 3) ( 3)x x xx x x
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 5
Example 4Simplify
2
2
3 25 6
x xx x
( 1)( 2)( 2)( 3)x xx x
13
xx
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 6
Rules of Multiplying Expressions
1. Take GCF of numerator(s) and denominator(s) if possible.
2. Factor the numerator(s) and denominator (s)
3. Cancel [up and down OR diagonally] out the factors if needed, multiply and simplify.
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 7
Example 5Simplify
2 23 24 6x yy x
4xy
22
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 8
Example 6Simplify 2 2
2
4 4 62 3 4
x x x xx x x
( 2)x
4 11 3( )
( )( )x x
x x
3 24
( )( )x xx
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 9
Undefined Values
Any number substituted into a variable which makes 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
01
0 10
undefined
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 10
Example 7Simplify . Identify any x-values for which the expression is
undefined.
2
3 43 4x
x x
1 4, ,1( 1) 3
xx
(3x + 4) (3x + 4)(x – 1)
3 4 0x 1 0x 43
x 1x
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 11
Example 8Simplify Identify any x-values for which the expression is undefined.
3 2 0x 2 3 0x 23
x 32
x
2
2
6 7 26 5 6x xx x
3 2 2 12 3 3 2x xx x
2 33 2
x x
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 12
Example 9
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.
–1(x2 – 4x)x2 – 2x – 8
–1(x)(x – 4)(x – 4)(x + 2)
Simplify.
2
2
42 8x x
x x
4 2x x
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 13
Rules of Divide Expressions1. Take GCF of numerator(s) and denominator(s)
if possible.
2. Take the reciprocal (flip) of the second fraction.
3. Factor the numerator(s) and denominators into simplest form
4. Cancel out the factors and cancel if needed,
5. Multiply and simplify
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 14
Example 103 12 6 242 4 4 8x xx x
3 42 2xx
4 26 4xx
2
2
1
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 15
Example 11Simplify
2 1
1
x
x
( 1)x
1 1
1x xx
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 16
Example 12Simplify
2 2
2
8 16 6 82 4
x x x xx x
( 4)( 2)( 2)
x xx
4 4
2x xx
2 22 4
x xx x
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 17
Your TurnSimplify and/or solve these problems
1. 2. 3.
4. 5.
5 4
3
13x yxy x y
2 2 63 5
x xx x
2 2
2 2
2 3 2 152 6
x x x xx x x x
4x2 – 92x + 3 = 5
6x – x2
x2 – 7x + 6
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 18
Example 13Simplify 2
( 4) 5 203 4 4 ( 2)(7 5)x x
x x x y
(7 5)5(3 2)yx
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 19
Example 14Solve. Check your solution and list restrictions of
9; 5x x
Note that x ≠ 5.(x + 5)(x – 5)(x – 5) = 14
x + 5 = 14x = 9
2 25 145
xx
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 20
Example 15Solve. Check your solution and list restrictions of
; 2No Solution x
2 3 10 72
x xx
05/03/23 16:19 10.3 Multiplying and Dividing Expressions 21
Assignment
Pg 580: 18 - 31
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