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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

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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

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Example 4Simplify

2

2

3 25 6

x xx x

( 1)( 2)( 2)( 3)x xx x

13

xx

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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.

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Example 5Simplify

2 23 24 6x yy x

4xy

22

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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

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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

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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

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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

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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

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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

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Example 103 12 6 242 4 4 8x xx x

3 42 2xx

4 26 4xx

2

2

1

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Example 11Simplify

2 1

1

x

x

( 1)x

1 1

1x xx

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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

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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

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Example 13Simplify 2

( 4) 5 203 4 4 ( 2)(7 5)x x

x x x y

(7 5)5(3 2)yx

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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

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Example 15Solve. Check your solution and list restrictions of

; 2No Solution x

2 3 10 72

x xx

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Assignment

Pg 580: 18 - 31