Download Unit 8 – Rational Expressions and Functions

Unit 8 – Rational Expressions and Functions
Unit 8.1 – Multiplying and Dividing Rational Expressions
A rational number is the quotient of two integers, with the denominator not 0.
A rational expression (or algebraic fraction) is the quotient of two polynomials, with the
denominator not 0.
x  a m  4 6 x 2  23x  15
are all rational expressions.
y
4
m4
4 y 3  26
P

Rational expressions are the elements of the set:  | P and Q are polynomials, with Q  0
Q

A rational function has the form f  x  
What is the domain of f  x  
P x 
where Q(x) ≠ 0.
Q x 
2
? Since x  5 cannot be 0, the domain is all real numbers,
x5
except 5.
Find the domain and write the domain in set notation:
3
3 x
8x  2
f x  
g x   2
h x  
7 x  14
3
x  4x  3
f x  
2
x 4
2
Writing rational expressions in lowest terms
Fundamental Property of Rational Numbers
a
a ac
If
is a rational number is if c is any nonzero real number, then 
. In words, the
b
b bc
numerator and denominator of a rational number may be multiplied or divided by the same
c
nonzero number without changing the value of the rational number. This is true because  1 .
c
To write a rational expression in lowest terms follow these steps:
1. Factor both numerator and denominator to find the GCF.
2. Apply the fundamental property.
Examples:
8k 8  k k
k

 1 
16 8  2 2
2
8k
cannot be simplified since the numerator cannot be factored.
16
 x  3
x 2  x  6  x  3 x  2   x  3


1 
2
x  5 x  6  x  3 x  2   x  3
 x  3
582722055
Page 1 of 8
Unit 8 – Rational Expressions and Functions
Examples:
x2  4
x3  27
2x  4
x 3
1  p3
1 p
3x  3 y  rx  ry
5 x  5 y  rx  ry
Simplifying rational expressions when numerator and denominator are opposites.
x 3
Example:
the numerator and denominator are opposites.
3 x
The denominator or numerator can be rewritten by multiplying it by -1.
x 3
1

 1
1 x  3 1
x 2  16  x  4  x  4 

4r
4 x
 x  4  x  4 

1  x  4 
x4
1
   x  4   x  4

Examples:
y2
2 y
Careful of the second one!
8b
8b
p2
4  p2
Multiplying Rational Expressions
1. Factor all numerators and denominators, completely.
2. Apply the fundamental property.
3. Multiply remaining factors, leaving in factored form.
4. Check to be sure product is in lowest terms.
5  x  1 3  x  x
5 x  5 3x 2



x 10 x  10
x
2  5  x  1

5   x  1  3  x  x
x  2  5   x  1
5 x  x  1 3  x
  

5 x  x  1 2
 1 1 1 
582722055
3x 3x

2
2
Page 2 of 8
Unit 8 – Rational Expressions and Functions
Examples:
2x  4
3x
6


5 x 5 x  10 25
2
x  2x x2  4x  4 x  2


x2  4
x2  x
x 1
2
x 3
x  25
1
 2

2
x  2 x  15 x  3 x  40 x  8
Dividing rational expressions.
Multiply the first (the dividend) by the reciprocal of the second (the divisor).
(Remember: to find the reciprocal of a rational expression, interchange the numerator and
denominator of the expression.)
Unit 8.2 – Adding and Subtracting Rational Expressions
Just like adding fractions (which are, after all, rational numbers), you need common
denominators to add rational expressions.
2 1
10 3

To add  , convert both fractions to a fraction with a common denominator
and add
3 5
15 15
13
numerators  . Simplify the result, if possible.
15
Rational expressions with a common denominator:
3y x 3y  x
 
5 5
5
m
p
m p
 2
 2
2
2
2
m p m p
m  p2
m p
In the second example we had to simplify.

 m  p  m  p 

1
m p
Rational expressions without a common denominator:
First write the expressions with a common denominator. One way to do this is to find the Least
Common Denominator (LCD).
Finding the Least Common Denominator
1. Factor each denominator.
2. Find the LCD. The LCD is the product of all different factors from each denominator, with
each factor raised to the greatest power in any denominator.
Finding common denominator examples (assume each is a denominator):
582722055
Page 3 of 8
Unit 8 – Rational Expressions and Functions
5 xy 2 and 2 x3 y : each is already factored. The different factors are 5, 2, x3, y2. So the LCD is
5  2  x3  y 2  10 x3 y 2
It is usually best to leave a common denominator in factored form…
x 2  2 x  8 and x 2  3x  2
x 2  2 x  8   x  4  x  2 
x 2  3x  2   x  2  x  1
LCD is  x  4  x  2  x  1
Examples:
x2  2 x  1
2
2 x  3x  2
x2  4 x  3
2
2 x  3x  1
4x  4
Add and subtract rational expressions with different denominators.
Write each rational expression with the LCD by multiplying its numerator and denominator by
the factors needed to get the LCD. (We are using the fundamental property of rational numbers,
i.e. multiplying the original rational expression by a form of 1)
Examples:
5
3
5 4
3 20 3 23

 the LCD is 8x




2x 8x
2x  4 8x 8x 8x 8x
6
5

What is the LCD? x  x  3
x x3
6  x  3
6
5
x 5



fundamental property
x x  3 x  x  3 x  x  3
582722055

6 x  18
5x

distributive and commutative properties
x  x  3 x  x  3

6 x  18  5 x
subtract the numerators
x  x  3

x  18
combine like terms in the numerator
x  x  3
Page 4 of 8
Unit 8 – Rational Expressions and Functions
CAREFUL: When subtracting rational expressions, the subtraction sign must be distributed to
every term in the numerator of the fraction that follows it.
7x
x  2 7 x   x  2


subtract the numerators
3x  1 3x  1
3x  1
7x  x  2

distributive property; watch your signs!
3x  1
6x  2

combine like terms in the numerator
3x  1
2  3 x  1

factor the numerator
3x  1
 2 lowest terms
Examples:
5 x  7  x  14

2x  7 2x  7
2
x

x  2 x 1
Recognize what to do when denominators are opposites. Multiply one by
1
. Now you’ll have
1
common denominators and can add or subtract.
x
8
x
8  1
x
8
x 8






x  2 2  x x  2  2  x  1 x  2 x  2 x  2
OR
x
8
x  1
8
x
8
8 x






x  2 2  x  x  2  1 2  x 2  x 2  x 2  x
These are equivalent expressions (how would you check this?)
Examples:
3
5
6
  2
(LCD is x  x  2  )
x  2 x x  2x
x4
2x  3
 2
(factor the denominators to determine LCD)
2
x  2 x  3 x  5x  6
Unit 8.3 – Complex Fractions
A complex fraction is an expression with a fraction in the numerator, the denominator, or both.
1
4
x2  9
1
x
x
x  1 examples of complex fractions.
x
x
2
2
6
2
2
x 1
582722055
Page 5 of 8
Unit 8 – Rational Expressions and Functions
Simplifying complex fractions:
If the numerator and denominator are already simplified, write the fraction as a division problem,
solve, and simplify.
If this is not so, there are two methods for simplifying complex fractions.
Method 1
1. Simplify the numerator and denominator separately.
2. Divide by multiplying the numerator by the reciprocal of the denominator.
3. Simplify the resulting fraction, if possible.
1 2x 1 2x 1

x  x x  x  2 x  1  3x  2  2 x  1  x  2 x  1
2 3x 2 3x  2
x
x
x 3x  2 3x  2
3

x
x x
x
Method 2
1. Multiply the numerator and denominator of the complex fraction by the LCD of the fractions
in the numerator and denominator.
2. Simplify the resulting fraction, if possible.
2
1

1
1
1
2
 2    x 2  x   x 2x 1
x


x 
x 1 
x 

2
2
2
2
3x  2

3
3
3    x 3 x   x

x
x
x
x

2
Which one to use? Doesn’t matter, use them both or pick the one you’re more comfortable with.
Simplifying a rational expression with negative exponents.
According to the rules of exponents, how would x 2 be simplified to an expression with only
1
positive exponents? 2
x
So for an expression such as
m1  p 2
, first write the expression with only positive exponents
2m2  p 1
and then simplify.
1 1

1
2
m p
m p2

2m 2  p 1 2  1  1
m2 p
1 1

m p2

and now simplify this rational expression
2 1

m2 p
582722055
Page 6 of 8
Unit 8 – Rational Expressions and Functions
Unit 8.4 – Equations with Rational Expressions and Graphs
Remember that we defined the domain of a rational expression as all possible values of the
independent or x variable. Any value that makes the denominator 0 is excluded from the domain.
The domain of a rational equation is the intersection of the domains of each rational expression
in the equation.
Find the domain of each equation:
2 3 7
 
figure the domain of each rational expression which are, in order
x 2 2x
x | x  0 , x | x  0 . The domain of the equation is the intersection of these three
domains or x | x  0
The essential fact here is the domain cannot contain a number that will make any of the rational
expressions in the original equation equal to 0.
Solve rational equations:
The easiest way to solve a rational equation is to multiply all terms in the equation by the LCD.
This will clear all the denominators. There is a caution, however. The resulting solutions may
not satisfy the original equation. You must either determine the original domain or check all
potential solutions in the original equation. I recommend you do both, just to make sure you
didn’t miss something.
2 3 7
 
the LCD is 2 x
x 2 2x
2 3
 7 
2 x     2 x   multiply both sides by the LCD
 x 2
 2x 
2
3
 7 
2 x    2 x    2 x   distributive property
 x
2
 2x 
4  3x  7 multiply
3x  3 subtract 4 from both sides
x  1 divide both sides by  3
582722055
Page 7 of 8
Unit 8 – Rational Expressions and Functions
Now you should check your answer in the original equation:
2 3 7
 
original equation
x 2 2x
2 3
7
 
substitute  1 for x
1 2 2  1
3
7

2
2
4 3
7
    convert  2 to fraction with common denominator
2 2
2
7
7
   true statement
2
2
2 
Graphs of rational functions.
A function defined by a rational expression is a rational function.
Because one or more values of x may be excluded from the domain of the function, the graph of
a rational function is often discontinuous. In other words, there will be one or more “breaks” in
the graph. These are typically vertical lines (called ‘vertical asymptotes’) and horizontal lines
(called ‘horizontal asymptotes’).
1
Example: f  x   has a vertical asymptote at x  0 and a horizontal asymptote at y  0 (since
x
there is no value of x that will make the function 0).
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
582722055
Page 8 of 8