Lesson 8-1: Multiplying and Dividing Rational Expressions

Rational Expression

Definition: a ratio of two polynomial expressions

x

x

13

8

To Simplify A Rational Expression

1. Make sure both the numerator and denominator are factored completely!!!

2. Look for common factors and cancel– Remember factors are things that are being

multiplied you can NEVER cancel things that are being added or subtracted!!!

3. Find out what conditions make the expression undefined and state them.

Examples: Simplify and state the values for x that result

in the expression being undefined

)128(6

632

yyy

yy 6412

8742

23

xx

xxx2.1.

Examples Cont… Simplify

wzz

zwz33

22

yxx

xxy223

3

baa

aba33

44

2

2

5.4.3.

Operations with Rational Expressions

To Multiply Rational Expressions:

Factor and cancel where possible. Then multiply numerators and denominators

To Divide Rational Expressions:

Rewrite the problem as a multiplication problem with the first expression times the reciprocal of the second expression. Then factor and cancel where possible. Multiply numerators and denominators

Define x-values for which the expression is undefined

Examples: Simplify

xy

b

yx

ab

10

9

25

1832

2

3

2

16

15

5

4

a

b

a

a

232

2

12

15

5

8

st

sr

r

st

83

3

37

4

35

21

15

14

zw

qp

zw

pq

7.

9.8.

6.

Polynomials in Numerator and Denominator

Rules are the same as before…1. Make sure everything is factored completely

2. Cancel common factors

3. Simplify and define x values for which the expression is undefined.

Examples: Simplify and define x values for which it is undefined

9

12

3

22

2

a

aa

a

a54

65

155

12

2

yy

yy

y

y10.11.

Examples:

34

1

1

32

2

kk

k

k

k23

3

2

6222

dd

d

dd

d13.12.

Simplifying complex fractions

A complex fraction is a rational expression whose numerator and/or denominator contains a rational expression

b

a

35

3

3

tt

t

tt

6

3 2

To simplify complex fractions

Same rules as before– Rewrite as multiplication with the reciprocal– Factor and cancel what you can– Simplify everything– Multiply to finish

Examples:

43163

2

2

xxxx14.

xyxyx

x

32

493

22

2

15.

Lesson 8-2: Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions

Finding Least Common Multiples and Least Common Denominators!– Use prime factorization

– Example: Find the LCM of 6 and 4 6 = 2·3 4 = 22

LCM= 22·3 = 12

Find the LCM

1. 18r2s5, 24r3st2, and 15s3t

2. 15a2bc3, 16b5c2, 20a3c6

3. a2 – 6a + 9 and a2 + a -12

4. 2k3 – 5k2 – 12k and k3 – 8k2 +16k

Add and Subtract Rational Expressions

Same as fractions…– To add two fractions we find the LCD, the same

things is going to happen with rational expressions

Examples: Simplify

27

1

9

8

abb

a

xy

y

y

x

1815

72

22

1

8

1

mnnm

5.

7.

6.

82

4

164

12

w

w

w

w

62

6

186

6

x

x

x

x

2012

1

583

12

x

x

xx

x8. 9. 10.

xy

xy11

11

x

yx1

1

11

11. 12.

xx

x4

42

14.

6

3x

x

xx

13.

Lesson 8-3: Graphing Rational Functions

Definitions

Continuity: graph may not be able to be traced without picking up pencil

Asymptote: a like that the graph of the function approaches, but never touches (this line is graphed as a dotted line)

Point discontinuity: a hole in the graph

Vertical Asymptote

How to find a Vertical Asymptote:

x = the value that makes the rational expression undefined

*Set the denominator of the rational expression equal to zero and solve.

Point Discontinuity

How to find point discontinuity:

* Factor completely

* Set any factor that cancels equal to zero and solve. Those are the

x values that are points of discontinuity

Graphing Rational Functions

f(x) = 1

3

x

f(x) =

Graphing Rational Functions

x

2

f(x) =

Graphing Rational Functions

3

12

x

x

f(x)=

Graphing Rational Functions

2)4(

2

xx

f(x) =

Graphing Rational Functions

3

92

x

x

f(x) =

Graphing Rational Functions

1

562

x

xx

Lesson 8-4: Direct, Joint, and Inverse Variation

Direct Variation

y varies directly as x if there is a nonzero constant, k, such that y = kx*k is called the constant of variation

1. Plug in the two values you have and solve for the missing variable

2. Plug in that variable and the other given value to solve for the requested answer

Example

If y varies directly as x and y = 12 when

x = -3, find y when x = 16.

Joint Variation

y varies jointly as x and z if there is a nonzero constant, k, such that y = kxz

* Follow the same directions as before

Example

Suppose y varies jointly as x and z. Find y when x = 8 and z = 3, if y = 16 when z = 2 and x = 5.

Inverse Variation

y varies inversely as x if there is a nonzero constant, k, such that xy = k or y=

kx

Example

If y varies inversely as x and y = 18 when x = -3, find y when x = -11

Lesson 8-6: Solving Rational Equations and Inequalities

Let’s review some old skills

How do you find the LCM of two monomials– 8x2y3 and 18x5

* Why do we find LCM’s with rational expressions?

Old Skills Cont…

What is it called when two fractions are equal to each other?

What process do we use to solve a problem like this?

To solve a rational equation:

1. Make sure the problem is written as a proportion

2. Cross Multiply

3. Solve for x

4. Check our answer

Examples

Solve

9

1

52

1x

102

3

3

2

x

Let’s put those old skills to new use…

Solve .

Solve . Check your solution.

3

3

158

10

5

22

2

xxx

xx

x

x

Examples

Solve

22

3

44

5

33

7

n

n

nn

n

Lesson 8-6: Day #2

Rational Inequalities

Solving Rational Inequalities

Step 1: State any excluded values (where the denominator of any fraction could equal zero)

Step 2: Solve the related equation

Step 3: Divide a numberline into intervals using answers from steps 1 and 2 to create intervals

Step 4: Test a value in each interval to determine which values satisfy the inequality

Examples

Solve

4

1

2

1

3

x

x

x

Examples

Solve

3

2

5

65xx

Examples

Solve

2

4

2

1

2

2

x

x

xx

x

Examples

Solve

2

3

1

1

2

xx

x