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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
