Chapter 12Chapter 12Final Exam ReviewFinal Exam Review

Section 12.4 Section 12.4 “Simplify Rational Expressions”“Simplify Rational Expressions”

A A RATIONAL EXPRESSIONRATIONAL EXPRESSION is is an expression that can be written as a an expression that can be written as a ratio (fraction) of two polynomials ratio (fraction) of two polynomials where the denominator is not 0.where the denominator is not 0.

3

2

x 58

272

ww

w

Simplify Rational ExpressionsSimplify Rational Expressions

To simplify a rational expression, you can To simplify a rational expression, you can factor the numerator and denominator factor the numerator and denominator and then divide out any common factors. and then divide out any common factors.

A rational expression is in A rational expression is in SIMPLEST SIMPLEST FORMFORM if the numerator and denominator if the numerator and denominator have no factors in common other than 1. have no factors in common other than 1.

34

12

x

xxxx

x

4

342

3

x

Simplify the rational expression, if possible. Simplify the rational expression, if possible.

86

1032

2

xx

xx)2)(4( xx

FactorFactor )2)(5( xx

4

5

x

x

103

452

2

zz

zz)2)(5( zz

FactorFactor

)1)(5( zz

2

1

z

z

Recognize Recognize oppositesopposites

)2)(5(

)1)(5(

zz

zz

Multiply by -1Multiply by -1Rewrite (z-5) as -(z+5)Rewrite (z-5) as -(z+5)

Section 12.5 Section 12.5 “Multiply and Divide Rational “Multiply and Divide Rational

Expressions”Expressions” Multiplying and dividing rational Multiplying and dividing rational

expressions is similar to multiplying and expressions is similar to multiplying and dividing fractions.dividing fractions.

Be sure to simplify your answer. Look to Be sure to simplify your answer. Look to cancel like terms when multiplying or cancel like terms when multiplying or dividing. dividing.

bd

ac

d

c

b

a

bc

ad

c

d

b

a

d

c

b

a

Multiply by reciprocal,Multiply by reciprocal,then look to cancel terms.then look to cancel terms.

5

33

8

15

5

2

y

y

y

y

r

r

5

12

3

16 2

Multiply by reciprocalMultiply by reciprocal

Find the product or quotient.Find the product or quotient.

EXAMPLE 1EXAMPLE 1

6

6

40

30

y

y 12

5

3

16 2 rr

4

3 36

80 3r

9

20 3r

xx

xx

xx

xx

2

2

2

2 34

36244

33

Find the product.Find the product.

EXAMPLE 2EXAMPLE 2

Multiply numerator and denominator.Multiply numerator and denominator.

))(36244(

)34)(33(22

22

xxxx

xxxx

)1()3)(3(4

)1)(3)(1(3

xxxx

xxxx Factor and look for common Factor and look for common factors to cancel.factors to cancel.

)3(4

)1(3

x

x

87

1

32

7722

2

xx

x

xx

xx

Find the quotient. Try it out.Find the quotient. Try it out.

EXAMPLE 2EXAMPLE 2

Multiply numerator Multiply numerator and denominator.and denominator.

)1)(32(

)87)(77(2

22

xxx

xxxx

)1)(1)(3(

)1)(8)(1(7

xxx

xxxxFactor and look for common Factor and look for common factors to cancel.factors to cancel.

)3(

)8(7

x

xx

1

87

32

77 2

2

2

x

xx

xx

xx Multiply by reciprocalMultiply by reciprocal

)3(65

52

xxx

x

Find the product. Find the product.

EXAMPLE 3EXAMPLE 3

Multiply numerator and denominator.Multiply numerator and denominator.

1

3

65

52

x

xx

x

65

)3(52

xx

xx Factor and look for common Factor and look for common factors to cancel.factors to cancel.

)2)(3(

)3(5

xx

xx

2

5

xx

Section 12.6 Section 12.6 “Add and Subtract Rational Expressions”“Add and Subtract Rational Expressions”

Adding and subtracting rational Adding and subtracting rational expressions is similar to adding and expressions is similar to adding and subtracting fractions.subtracting fractions.

Be sure to simplify your answer. Be sure to simplify your answer.

c

ba

c

b

c

a

c

ba

c

b

c

a

Denominator must be COMMON!!!Denominator must be COMMON!!!

82

1

103

422

xx

x

xx

x

Find the sum or difference.Find the sum or difference. EXAMPLE 5EXAMPLE 5

)2)(5(

4

xx

x

LCD = (x+5)(x – 2)(x + 4)LCD = (x+5)(x – 2)(x + 4)

)4(

)4(

x

x

)4)(2(

1

xx

x

)5(

)5(

x

x

)4)(2)(5(

)5)(1()4)(4(

xxx

xxxx

)4)(2)(5(

)54(168 22

xxx

xxxx

)4)(2(

1

)2)(5(

4

xx

x

xx

x

)4)(2)(5(

214

xxx

x

Section 12.7 Section 12.7 “Solve Rational Equations”“Solve Rational Equations”

A A RATIONAL EQUATIONRATIONAL EQUATION is an is an equation that contains one or more equation that contains one or more rational expressions. rational expressions.

One method for solving a rational equation One method for solving a rational equation is to use the cross products property. (You is to use the cross products property. (You can use this method when both sides of the can use this method when both sides of the equation are single rational expressions). equation are single rational expressions).

24

6 x

x

32

5 y

y

yy 215 2

Cross multiplyCross multiply

Solve the equation. Check your solution.Solve the equation. Check your solution.

)2()3)(5( yy

1520 2 yy

3;5

)3)(5(0

y

yy

Check for extraneous solutions

32

5 y

y

3

3

5

53

3

23

5

32

5 y

y

3

5

3

53

5

25

5

y = 5y = 5 y = -3y = -3