Intermediate Algebra Unit 6 Rational Expressions

Intermediate Algebra

Exam 2 Material

Rational Expressions

Rational Expression

• A ratio of two polynomials where the denominator is not zero (an “ugly fraction” with a variable in a denominator)

• Example:

• Will the value of the denominator ever be zero?If x = - 3, then the denominator becomes 0, so we say that – 3 is a restricted value of x

• What is the “domain” of the rational expression (all acceptable values of the variable)?Domain is the set of all real numbers except - 3.Domain: {x | x ≠ -3}

3

22

x

xx

Finding Restricted Values and Domains of Rational Expressions

• Completely factor the denominator

• Make equations by setting each factor of the denominator equal to zero

• Solve the equations to find restricted values

• The domain will be all real numbers that are not restricted

Example

Find the domain:

• Factor the denominator:(x – 2)(x + 2)

• Set each factor equal to zero and solve the equations:x – 2 = 0 and x + 2 = 0x = 2 and x = -2 (Restricted Values)

• Domain:{x | x ≠ -2, x ≠ 2}

4

532

x

x

Evaluating Rational Expressions

• To “evaluate” a rational expression means to find its “value” when variables are replaced by specific “unrestricted” numbers inside parentheses

• Example: 2for 32

32 Evaluate

2 -x

xx

x

32

322 22

2

344

34

3

7

3

7

Fundamental Principle of Fractions

• If the numerator and denominator of a fraction contain a common factor, that factor may be divided out to reduce the fraction to lowest terms:

ac

ab

c

b

18

12

332

322

3

2

place.each in left is "1" out, divided are factorscommon When

x5

5

x

1

1

1

1

1

1

1

1

1

Reducing Rational Expressions to Lowest Terms

• Completely factor both numerator and denominator

• Apply the fundamental principle of fractions: divide out common factors that are found in both the numerator and the denominator

Example of Reducing Rational Expressions to Lowest Terms

• Reduce to lowest terms:

• Factor top and bottom:

• Divide out common factors to get:

63

243 3

x

x

23

83 3

x

x

23

4223 2

x

xxx

422 xx

1

11

1

Example of Reducing Rational Expressions to Lowest Terms

• Reduce to lowest terms:

• Factor top and bottom:

• Divide out common factors to get:

x

x

3

3

31

31

x

x

31

31

x

x

1

3

3

x

x

1

Equivalent Formsof Rational Expressions

• All of the following are equivalent:

• In words this would say that a negative factor in the numerator or denominator can be moved, respectively, to a negative factor in the denominator or numerator, or can be moved to the front of the fraction, or vice versa

q

p

q

p

q

p

Example of Using Equivalent Forms of Rational Expressions

• Write equivalent forms of:

2

5

x

x

2

5

x

x

2

5

x

x

2

5

x

x

2

5

x

x

2

5

x

x

Homework Problems

• Section: 6.1

• Page: 401

• Problems: Odd: 3 – 9, 13 – 23, 27 – 63, 67 – 73

• MyMathLab Homework Assignment 6.1 for practice

• MyMathLab Quiz 6.1 for grade

Multiplying Rational Expressions (Same as Multiplying Fractions)

• Factor each numerator and denominator• Divide out common factors • Write answer (leave polynomials in factored

form)• Example:

28

15

9

41 1

1

1

1121

5

722

53

33

22

Example of MultiplyingRational Expressions

Completely factor each top and bottom:

Divide out common factors:

43

23

8143

8232

2

x

x

xx

xx

43

23

423

243

x

x

xx

xx

1

1

1

1

4

2

x

x

Dividing Rational Expressions(Same as Dividing Fractions)

• Invert the divisor and change problem to multiplication

• Example:

d

c

b

a

c

d

b

a

4

3

3

2

bc

ad

3

4

3

2

9

8

Example of DividingRational Expressions

27

48

9

2 352 yyy

35

2

48

27

9

2

yy

y

124

27

9

223

2

yy

y

122

32 yy

3

21

1 1

y

Homework Problems

• Section: 6.2

• Page: 408

• Problems: Odd: 3 – 25, 29 – 61



Finding the Least Common Denominator, LCD, of Rational

Expressions

• Completely factor each denominator

• Construct the LCD by writing down each factor the maximum number of times it is found in any denominator

Example of Finding the LCD

• Given three denominators, find the LCD:, ,

• Factor each denominator:

• Construct LCD by writing each factor the maximum number of times it’s found in any denominator:

123 2 x

123 2x

16164 2 xx

16164 2 xx

126 x

126x

43 2x 223 xx

26 x

444 2 xx 2222 xx

232 x

LCD

LCD

222322 xxx

2212 2 xx

Equivalent Fractions

• The fundamental principle of fractions, mentioned earlier, says:

• In words, this says that when numerator and denominator of a fraction are multiplied by the same factor, the result is equivalent to the original fraction

.

ac

ab

c

b

18

12

36

263

2

Writing Equivalent Fractions With Specified Denominator

• Given a fraction and a desired denominator for an equivalent fraction that is a multiple of the original denominator, write an equivalent fraction by multiplying both the numerator and denominator of the original fraction by all factors of the desired denominator not found in the original denominator

• To accomplish this goal, it is usually best to completely factor both the original denominator and the desired denominator

24

20

Example

Write an equivalent fraction to the given fraction that has a denominator of 24:

622

522

6

5

6

524

? 326 322224

:rdenominatoeach Factor

Example

Write an equivalent rational expression to the given one that has a denominator of :

yyyyy

y

242

?2232

12 yyyy

112242 23 yyyyyy

yyy 242 23

1

222 yy

y

yy

y

121

122

yyy

yy

121

22 2

yyy

yy

yyy

yy

242

42223

2

:rdenominatoeach Factor

Homework Problems

• Section: 6.3

• Page: 414

• Problems: Odd: 5 – 43, 51 – 69



Adding and Subtracting Rational Expressions (Same as Fractions)• Find a least common denominator, LCD,

for the rational expressions• Write each fraction as an equivalent

fraction having the LCD• Write the answer by adding or

subtracting numerators as indicated, and keeping the LCD

• If possible, reduce the answer to lowest terms

Example

• Find a least common denominator, LCD, for the rational expressions:

• Write each fraction as an equivalent fraction having the LCD:

• Write the answer by adding or subtracting numerators as indicated, and keeping the LCD:

• If possible, reduce the answer to lowest terms

yyy

y

yy

y 1

242

3222

y

1yy 112 yy 112 yyy

LCD

112

1112

112

3

112

122

yyy

yy

yyy

yy

yyy

yy

yyy

y

yy

y 1

112

3

1

2

112

122322 222

yyy

yyyyy

112

2423422 222

yyy

yyyyy

112

222

yyy

yy reduce!t on'fraction w factor,t won' topSince

Homework Problems

• Section: 6.4

• Page: 422

• Problems: Odd: 9 – 21, 25 – 47, 51 – 71



Complex Fraction

• A “fraction” that contains a rational expression in its numerator, or in its denominator, or both

• Example:

• Think of it as “fractions inside of a fraction”• Every complex fraction can be simplified to a

rational expression (ratio of two polynomials)

y

x

65

231

Two Methods for Simplifying Complex Fractions

• Method One– Do math on top to get a single fraction– Do math on bottom to get a single fraction– Divide top fraction by bottom fraction

• Method Two (Usually preferred)– Find the LCD of all of the “little fractions”– Multiply the complex fraction by “1” where “1”

is the LCD of the little fractions over itself

Method One Example ofSimplifying a Complex Fraction

• Do math on top to get single fraction:

• Do math on bottom to get single fraction:

• Top fraction divided by bottom:

y

x

65

231

x

xyy

5

122

23

1

x

1

2

3

1

x

x

x

x 3

6

3

1

x

x

3

61

y6

5 :fraction singlealready is bottom case, In this

yx

x

6

5

3

61

5

6

3

61 y

x

x2

Method Two Example ofSimplifying a Complex Fraction

• Find the LCD of all of the “little fractions”:

• Multiply the complex fraction by “1” where “1” is the LCD of the little fractions over itself

y

x

65

231

xy6

y

x

65

231

x

xyy

5

122

yxy

xyxxy

630

112

36

161

6

xy

xy

Homework Problems

• Section: 6.5

• Page: 431

• Problems: Odd: 7 – 35



Other Types of Equations

• Thus far techniques have been discussed for solving all linear and some quadratic equations

• Now address techniques for identifying and solving “rational equations”

Rational Equations

• Technical Definition: An equation that contains a rational expression

• Practical Definition: An equation that has a variable in a denominator

• Example:

3

2

1

5

32

12

xxxx

Solving Rational Equations

1. Find “restricted values” for the equation by setting every denominator that contains a variable equal to zero and solving

2. Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions

3. Solve the resulting equation to find apparent solutions

4. Solutions are all apparent solutions that are not restricted

Example

3

2

1

5

32

12

xxxx

RV

01x 03 x

0322 xx

031 xx

01x

03 x

OR1x 3x

SolvedAlready

SolvedAlready 3

2

1

5

31

1

xxxx

LCD 31 xx 1

LCD

3

2

1

5

31

1

xxxx

12351 xx221551 xx

1731 x

x316

3

16x RV!Not

Example

1

1

2

1

1

22

mm

RV

01m 01m

012 m

011 mm

01m

OR1m 1m

SolvedAlready 1

1

2

1

11

2

mmm

LCD 112 mm

12114 mmm

11

1

2

1

11

2 LCD

mmm

2214 2 mm2214 2 mm

320 2 mm

130 mm

01 03 morm1 3 morm

Formula

• Any equation containing more than one variable

• To solve a formula for a specific variable we must use appropriate techniques to isolate that variable on one side of the equal sign

• The technique we use in solving depends on the type of equation for the variable for which we are solving

Example of Different Types of Equations for the Same Formula

• Consider the formula:

• What type of equation for A?Linear (variable to first power)

• What type of equation for B?Quadratic (variable to second power)

• What type of equation for C?Rational (variable in denominator)

1

43

2

C

BA

Solving Formulas Involving Rational Equations

• Use the steps previously discussed for solving rational equations:

1. Find “restricted values” for the equation by setting every denominator that contains the variable being solved for equal to zero and solving

2. Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions

3. Solve the resulting equation to find apparent solutions

4. Solutions are all apparent solutions that are not restricted

Solve the Formula for C:

Since the formula is rational for C, find RV:

Multiply both sides by LCD:

1

43

2

C

BA

01C 1C

1C

11

431

2

CC

BAC

2433 BACAC

Example Continued

Solve resulting equation and check apparent answer with RV:

2433 BACAC 343 2 ABCAC

343 2 ABCA

3

34

3

3 2

A

AB

A

CA

3

34 2

A

ABC RVNot

Cfor linear Now

Homework Problems

• Section: 6.6

• Page: 439

• Problems: Odd: 17 – 69, 73 – 87


• ( No MyMathLab Quiz until we finish Section 6.7 )

Applications of Rational Expressions

• Word problems that translate to rational expressions are handled the same as all other word problems

• On the next slide we give an example of such a problem

Example

When three more than a number is divided by twice the number, the result is the same as the original number. Find all numbers that satisfy these conditions.

.

:Unknownsnumber The x

xx

x

2

3

:RV02 x0x

xxx

xx 2

2

32

223 xx

320 2 xx

1320 xx

01or 032 xx

1or 32 xx

1or 2

3 xx

Homework Problems

• Section: 6.7

• Page: 449

• Problems: Odd: 3 –9


• MyMathLab Quiz 6.6 - 6.7 for grade

Intermediate Algebra Unit 6 Rational Expressions

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