6.1 1

Section 6.1 Rational Expression & Functions:Definitions, Multiplying, Dividing

Fractions - a Quick Review Definitions:

Rational Functions, Expressions Finding the Domains (and Exclusions) of

Rational Functions Simplifying Rational Functions Simplifying by factoring out -1

6.1 2

Fractions - Review Q: When can you add or subtract fractions?

A: Only when denominators are the same Q: What do you do when denominators are not the same?

A: Use their LCD to create equivalent fractions. Q: How do you multiply fractions?

A: Factor all tops and factor all bottoms, cancel matching factors, multiply tops and bottoms

Q: What do you do first when dividing fractions? A: Turn division into multiplication : reciprocal the divisor.

Rational Expressions are Polynomial Fractions ! Same rules!

6.1 3

Definitions

6.1 4

Finding the Domain (and exclusions) of a Rational Function

Recall the domain of a function is the set ofall real numbers for which the function is defined. - What real values make this function undefined

(divided by 0)?

Factor: x2 + 2x – 24 = (x – 4)(x + 6) {x | x is Real, except for 4 or -6}

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Graphs of Rational Functions

2t + 5 ≠ 0

2t ≠ -5

t ≠ -5/2

t=-5/2 is an Asymptote

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Definitions Horizontal Asymptote – A horizontal line that

the graph of a function approaches as x values get very large or very small.

Vertical Asymptote – A vertical line that the graph of a function approaches as x values approach a fixed number

6.1 7

More Properties of Fractions - Review

6.1 8

2

3

22

23

4

24 4)3(

)4)(3(3

12ba

babbaa

abba

Simplifying Rational Expressions(In general, the expressions are NOT equivalent)

3)3(

)3)(3(392

x

xxx

xx

6.1 9

5)5()3(

)5)(5()3)(5(

25152

2

2

xx

xxx

xxx

xx

First Factor and Identify domain exclusions,Then Simplify

33

1)93)(3(

)93(27

932

2

3

2

x

xxxxxx

xxx

3,22

2)2)(2)(3(

)2)(3(21243

122223

2

xxxxx

xxxxx

xx

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Multiplying Fractions

(First find domain exclusions)Factor expressions,

then cancel like factors

3,0)3()3(

)3(3

96 23232

aaaaa

aaaa

aaa

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Example – Step by Step

4)3(

)3)(20()5)(3)(3(

)3)(20()5)(96(

35

2096 22222

xx

xxxxx

xxxxx

xx

xxx

1

4 1 1

1 x

1. Write down original problem2. Combine with parentheses3. Find any polynomials that need factoring4. Rewrite (if any factoring was done)5. Identify domain exclusions6. Cancel out matching factors7. Simplify the answer

x≠0,3

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Board Practice – Rational Multiplication

1. Write original problem2. Combine w/ parens3. Factor polynomials4. Rewrite (if any factoring)5. Identify domain exclusions6. Cancel matching factors7. Simplify the answer

6456

4956

2

2

2

2

aaa

aaa

bxxbxxxx

22)2(

22

5

483 3m

bamba

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Finding Powers of Rational Expressions Factor and Simplify (if possible) before applying the power If part of a larger expression, see if any terms cancel out Multiply out the terms in the numerator,

multiply out the terms in the denominator. Leave in simplified factored form

22

222

2 )6()5(

)6()6()5)(5(

)6(5

65

xxx

xxxxxx

xxx

xxx

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Dividing FractionsChange Divide to

Multiply by Reciprocal,follow multiply procedure

23

4213

1422

4233

18

3342

18

2

2

2

23

2

23

xxxx

xxx

xxxxx

xxx

xxx

xxx

x

6.1 15

Board Practice - Rational Division

2242

448

2

23

xxx

xx

)2(143

b

xbb

5

483 3m

bamba

1. Write original problem2. Combine w/ parens3. Factor polynomials4. Rewrite (if any factoring)5. Identify domain exclusions6. Cancel matching factors7. Simplify the answer

6.1 16

Mixed Operations Multiplications & Division are done left to right In effect, make each divisor into a reciprocal

22

)2)(3()2)(3(

)2(1

)3(1)6()2()3()6( 22

xx

xxxx

xxxxxxxx

6.1 17

What Next? Present Section 6.2 Add/Subtract Rational Expressions