Transcript 6.1 The Fundamental Property of Rational Expressions • • Rational Expression – has the form: P Q where P and Q are polynomials with Q not equal to.

6.1 The Fundamental Property of
Rational Expressions
•
•
Rational Expression – has the form:
P
Q
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.
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:
PK P

QK Q
6.1 The Fundamental Property of
Rational Expressions
•
Example: Find where the following p  5
rational expression is undefined:
3p  2
1. Set the denominator equal to zero. 3 p  2  0
-2
2. Solve: 3 p  2  p 
3
3. The expression is undefined for:
-2
p
3
6.1 The Fundamental Property of
Rational Expressions
•
Example: Write the rational
expression in lowest terms:
3x  12
5 x  20
3x  12 3( x  4)

5 x  20 5( x  4)
3
(
x

4
)
3
2. By the fundamental property:

1. Factor:
5( x  4)
3. The expression is undefined for: x  4
5
6.2 Multiplying and Dividing
Rational Expressions
• Multiplying Rational Expressions – product
of two rational expressions is given by:
P R
PR


Q S
QS
• Dividing Rational Expressions – quotient of
two rational expressions is given by:
P R
P S
PS




Q S
Q R
QR
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:
x 2  3x x 2  5 x  4
 2
2
x  3x  4 x  2 x  3
x( x  3)
( x  4)(x  1)

( x  4)(x  1) ( x  1)(x  3)
• Cancel to get in lowest terms:
x
x 1
6.2 Multiplying and Dividing
Rational Expressions
x2  4
( x  2)(x  3)
• Example - divide:

( x  3)(x  2)
2x
x2  4
2x


( x  3)(x  2) ( x  2)(x  3)
• Factor:
( x  2)(x  2)
2x


( x  3)(x  2) ( x  2)(x  3)
• Cancel to get in lowest terms:
2x
( x  3) 2
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
•
5
3
and 3
Find the LCD for:
2
6r
4r
1. Factor both denominators
6r  2  3  r
2
4r  2  r
3
2
2. The LCD is the product of the
largest power of each factor: 22  3  r 3
2
3
 12r
3
6.3 Least Common Denominators
•
Rewrite the expression 12 p
 3
2
with the given
p  8 p p  4 p 2  32 p
denominator:
2
p
 8 p  p( p  8)
1. Factor both
denominators: p 3  4 p 2  32 p  p( p  8)( p  4)
2. Multiply top and
bottom by (p – 4)
12 p
p4
12 p( p  4)


p( p  8) p  4 p( p  8)( p  4)
12 p 2  48 p
 3
p  4 p 2  32 p
6.4 Adding and Subtracting Rational
Expressions
• Adding Rational Expressions:
P
If
and R are rational expressions, then
Q
Q
P
Q
 
R
Q
P R
Q
• Subtracting Rational Expressions:
If P and R are rational expressions, then
Q
Q
P
Q
 QR  PQ R
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:
2x
1

2
x 1 x 1
1. Factor denominators
to get the LCD:
2. Multiply to get a
common denominator:
2x
1

( x  1)(x  1) x  1
LCD is ( x  1)(x  1)
2x
1 x 1


( x  1)(x  1) x  1 x  1
2x
 x 1


( x  1)(x  1) ( x  1)(x  1)

3. Add and
2x  x  1
x 1
1
simplify:  ( x  1)(x  1)  ( x  1)(x  1)  ( x  1)
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 fraction
2. Change the complex fraction to a division
problem
3. Perform the division by multiplying by the
reciprocal
6.5 Complex Fractions
3
6

x
• Example:
x
1

4
8
1. Write top and 6  x  3
x
x
bottom as a

single fraction 4x  22  18
2. Change to
division problem
6 x 3

x
3. Multiply by the
6 x 3
8
reciprocal and


x
2 x 1
simplify
6x
x
2x
8
 3x

 18

2 x 1
8

3( 2 x 1)
x
6 x 3
x
2 x 1
8
 2 x81  24x
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
•
3
6

x
Example:
x
1

4
8
1. Find the LCD: the denominators are 4, 8, and x
so the LCD is 8x.
3
3


8
x
6

8
x
(
6
)

8
x
(
)
2. Multiply top
x
x


x
1
and bottom
8x 4  8  8x( 4x )  8x( 18 )
by this LCD.
3. Simplify:  48x  24  24(2 x  1)  24
2
2x  x
x(2 x  1)
x
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:
2
1
 2
2
x  x x 1
1. Factor to get LCD
LCD = x(x - 1)(x + 1)
2
1

x( x  1) ( x  1)(x  1)
2. Multiply both 2 x( x  1)(x  1) x( x  1)(x  1)

sides by LCD
x( x  1)
( x  1)(x  1)
2( x  1)  x
6.6 Solving Equations Involving
Rational Expressions
2
1
• Example (continued):
 2
2
x  x x 1
3. Solve the equation 2( x  1)  x
2x  2  x
x  2
4. Check solution
2
1

2
- 2  - 2 - 22  1
2 1

6 3
6.7 Applications of Rational
Expressions
• Distance, Rate, and time:
d  rt , r  dt , and t  dr
• Rate of Work - If one job can be completed in t units of
time, then the rate of work is:
1
r  job per unit time
t
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  x  2
5 x 3
2 x
2
3(5  x)
 3(5  x)
2. Multiply by
5 x
3
LCD: 3(5+x)
3(2  x)  2(5  x)
6  3 x  10  2 x
3. Subtract 2x and 6
x4
6.7 Applications of Rational Expressions
•
1.
2.
3.
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?
Table:
Rate Time
Part of Job Done
Regular
1/6
x
x/6
Substitute
1/8
x
x/8
x x
 1
6 8
Multiply by LCD: 24 4 x  3 x  24
Equation:
7 x  24
4.
Solve:
x  247 hours