6.1 The Fundamental Property of Rational Expressions Rational Expression – has the form: where P...

6.1 The Fundamental Property of Rational Expressions

• Rational Expression – has the form:

where P and Q are polynomials with Q not equal to zero.

• Determining when a rational expression is undefined:

1. Set the denominator equal to zero.2. Solve the resulting equation.3. The solutions are points where the rational expression

is undefined.

Q

P

6.1 The Fundamental Property of Rational Expressions

• Lowest terms – A rational expression P/Q is in lowest terms if the greatest common factor of the numerator and the denominator is 1.

• Fundamental property of rational expressions – If P/Q is a rational expression and if K represents any polynomial where K 0, then:

Q

P

QK

PK

6.1 The Fundamental Property of Rational Expressions

• Example: Find where the following rational expression is undefined:

1. Set the denominator equal to zero.

2. Solve:

3. The expression is undefined for:

23

5

p

p

3

2-p

023 p

3

223

-p p

6.1 The Fundamental Property of Rational Expressions

• Example: Write the rational expression in lowest terms:

1. Factor:

2. By the fundamental property:

3. The expression is undefined for:

205

123

x

x

4x

)4(5

)4(3

205

123

x

x

x

x

5

3

)4(5

4(3

x

)x

6.2 Multiplying and Dividing Rational Expressions

• Multiplying Rational Expressions – product of two rational expressions is given by:

• Dividing Rational Expressions – quotient of two rational expressions is given by:

QS

PR

S

R

Q

P

QR

PS

R

S

Q

P

S

R

Q

P

6.2 Multiplying and Dividing Rational Expressions

• Multiplying or Dividing Rational Expressions:

1. Factor completely

2. Multiply (multiply by reciprocal for division)

3. Write in lowest terms using the fundamental property

6.2 Multiplying and Dividing Rational Expressions

• Example - multiply:

• Factor:

• Cancel to get in lowest terms:1x

x

32

45

43

32

2

2

2

xx

xx

xx

xx

)3)(1(

)1)(4(

)1)(4(

)3(

xx

xx

xx

xx

6.2 Multiplying and Dividing Rational Expressions

• Example - divide:

• Factor:

• Cancel to get in lowest terms:

2)3(

2

x

x

)3)(2(

2

)2)(3(

4

2

)3)(2(

)2)(3(

4

2

2

xx

x

xx

x

x

xx

xx

x

)3)(2(

2

)2)(3(

)2)(2(

xx

x

xx

xx

6.3 Least Common Denominators

• Finding the least common denominator for rational expressions:

1. Factor each denominator

2. List the factors using the maximum number of times each one occurs

3. Multiply the factors from step 2 to get the least common denominator

6.3 Least Common Denominators

• Find the LCD for:

1. Factor both denominators

2. The LCD is the product of the largest power of each factor: 332 1232 rr

32 4

3 and

6

5

rr

323

22

24

326

rr

rr

6.3 Least Common Denominators

• Rewrite the expression with the given denominator:

1. Factor both denominators:

2. Multiply top and bottom by (p – 4)

ppppp

p

3248

12232

)4)(8(324

)8(823

2

pppppp

pppp

ppp

pp

ppp

pp

p

p

pp

p

324

4812

)4)(8(

)4(12

4

4

)8(

12

23

2

6.4 Adding and Subtracting Rational Expressions

• Adding Rational Expressions:If and are rational expressions, then

• Subtracting Rational Expressions:If and are rational expressions, then

QRP

QR

QP

QRP

QR

QP

QP

QR

QP

QR

6.4 Adding and Subtracting Rational Expressions

• Adding/Subtracting when the denominators are different rational expressions:

1. Find the LCD

2. Rewrite fractions – multiply top and bottom of each to get the LCD in the denominator

3. Add the numerators (the LCD is the denominator

4. Write in lowest terms

6.4 Adding/Subtracting Rational Expressions

• Add:1. Factor denominators

to get the LCD:

2. Multiply to get acommon denominator:

3. Add andsimplify:

1

1

1

22

xx

x

)1)(1( is LCD

1

1

)1)(1(

2

xx

xxx

x

)1)(1(

1

)1)(1(

2

1

1

1

1

)1)(1(

2

xx

x

xx

x

x

x

xxx

x

)1(

1

)1)(1(

1

)1)(1(

12

xxx

x

xx

xx

6.5 Complex Fractions

• Complex Fraction – a rational expression with fractions in the numerator, denominator or both

• To simplify a complex fraction (method 1):1. Write both the numerator and denominator as

a single fraction2. Change the complex fraction to a division

problem3. Perform the division by multiplying by the

reciprocal

6.5 Complex Fractions• Example:

1. Write top and bottom as a single fraction

2. Change to division problem

3. Multiply by thereciprocal and simplify

81

4

36

xx

812

36

81

82

36

81

22

4

36

xx

x

xxx

x

xxx

x

81236 x

xx

xxxx

xxx 24

128)12(3

12836

6.5 Complex Fractions

• To simplify a complex fraction (method 2):

1. Find the LCD of all fractions within the complex fraction

2. Multiply both the numerator and the denominator of the complex fraction by this LCD. Write your answer in lowest terms

6.5 Complex Fractions• Example:

1. Find the LCD: the denominators are 4, 8, and x so the LCD is 8x.

2. Multiply top and bottom by this LCD.

3. Simplify:

81

4

36

xx

)(8)(8

)(8)6(8

8

68

81

4

3

81

4

3

xx

xx

x

xx

xx

x

xxx

x

xx

x 24

)12(

)12(24

2

24482

6.6 Solving Equations Involving Rational Expressions

1. Multiply both sides of the equation by the LCD

2. Solve the resulting equation

3. Check each solution you get – reject any answer that causes a denominator to equal zero.

6.6 Solving Equations Involving Rational Expressions

• Solve:

1. Factor to get LCDLCD = x(x - 1)(x + 1)

2. Multiply both sides by LCD

1

1

x

222

xx

)1)(1(

1

)1(

2

xxxx

)1)(1(

)1)(1(

)1(

)1)(1(2

xx

xxx

xx

xxx

xx )1(2

6.6 Solving Equations Involving Rational Expressions

• Example (continued):

3. Solve the equation

4. Check solution

1

1

x

222

xx

2

22

)1(2

x

xx

xx

3

1

6

2

12-

1

2-2-

222

6.7 Applications of Rational Expressions

• Distance, Rate, and time:

• Rate of Work - If one job can be completed in t units of time, then the rate of work is:

rd

td trrtd and , ,

unit timeper job 1

tr

6.7 Applications of Rational Expressions• Example: If the same number is added to the

numerator and the denominator of the fraction 2/5, the result is 2/3. What is the number?

1. Equation

2. Multiply by LCD: 3(5+x)

3. Subtract 2x and 6 4x

3

2

5

2

x

x

xx

xx

xx

xx

21036

)5(2)2(33

2)5(3

5

2)5(3

6.7 Applications of Rational Expressions

• Example: It takes a mail carrier 6 hr to cover her route. It takes a substitute 8 hr. How long does it take if they work together?

1. Table:

2. Equation:

3. Multiply by LCD: 24

4. Solve:

Rate Time Part of Job Done

Regular 1/6 x x/6

Substitute 1/8 x x/8

hours 724x

186

xx

247

2434

x

xx