R.6
Rational Section Expressions
A rational expression is a fraction of two polynomials.
1
2 , x2 .
For example: 2 , x ,
2
3
x 5
x 6
W hen working with rational expressions we usually are
concerned with 4 things.
1. Determine the domain of the expression
2. Simplify it or reduce it to its lowest terms
3. Multiply, divide, add, subtract rational expressions
4. Simplify complex rational expressions
The Domain of a Rational Expression
A very important, but often overlooked, mathematical
principle is that you cannot divided by zero. A rational
expression is really a division problem. It is the
numerator divided by the denominator.
For example, x 5 is really x 5 divided by x 3 .
x 3
If you cannot divide by zero, then the divisor of this
problem, namely, x 3 cannot be zero. For the domain,
x can be any number except when x 3 0 or x 3.
Therefore the domain of x 5 is {x | x 3}.
x 3
1
Math 1513 Sec R.6
Find the domain of x 2 .
2
x 4
Simplifying/Reducing Rational Expressions
It might be helpful to review how you reduce simple
fractions. For example, how do you reduce 20 ?
25
One way to do it would be: 20 45 4 5 4
25
55
5
5 5
The same technique can be applied to more complicated
2
rational expressions. For example, reduce: 2 x 6 .
x x6
First factor the numerator and denominator.
2x 6 2( x 3)
x2 x 6 ( x 3)( x 2)
You can cancel common factors in the numerator and
denominator.
2( x 3) 2 ( x 3) 2
( x 3)( x 2)
( x 2)
( x 3) ( x 2)
2
Math 1513 Sec R.6
Very Important Note on Canceling
The only time you can cancel a term in the numerator
and denominator is when the term is multiplied by
everything on the top and bottom. That is, it is a
factor of the numerator and denominator. Before you
draw slashes through terms, ask yourself the
question, are these factors?
I saw a cartoon once where a trigonometry student
claimed that sin x 6 . He arrived at this answer by
n
some very innovative and very wrong canceling:
sin x sin x six
n
n
Multiplying Rational Expressions
It might be helpful to review how you multiply simple
fractions. For example, how do you multiply 2 3 ?
57
The rule is to multiply the numerators and multiply the
denominators.
2 3 23 6
5 7 5 7
35
The same technique can be applied to more complicated
rational expressions. For example, multiply: x 6 x 3 .
x2 x4
x 6 x 3 ( x 6)( x 3) 3x x2 18
x 2 x 4 ( x 2)( x 4)
x2 6 x 8
3
Math 1513 Sec R.6
Dividing Rational Expressions
Dividing rational expressions is very similar to multiplying
them. The only difference is that you invert the divisor
first and turn the problem into multiplication.
It might be helpful to review how you divide simple
fractions. For example, how do you divide 2 3 ?
57
The rule is to invert the divisor and then multiply.
2 3 2 7 27 14
5 7 5 3 53 15
The same technique can be applied to more complicated
rational expressions. For example, divide: x 6 x 3 .
x2
x4
x 6 x 3 x 6 x 4 ( x 6)( x 4)
x 2 x 4 x 2 x 3 ( x 2)( x 3)
4
Math 1513 Sec R.6
Adding and Subtracting Rational Expressions
It might be helpful to review how you add or subtract
simple fractions. For example, how do you add 2 3 ?
57
The rule is to
1. Get a least common denominator (LCD). This consists
of the smallest product that contains all the factors of
the denominators.
In our example the LCD would be 57 .
2. Convert each fraction to an equivalent fraction with the
LCD. In our example this would look like:
23
57
57 57
3. Add the numerators, put the answer over the LCD,
and reduce, if possible.
5
Math 1513 Sec R.6
The same technique can be applied to more complicated
rational expressions.
For example, add: x 1 x 1 .
2
3x 6
x 4
Factor the denominators to get the LCD:
x 1 x 1
2
3x 6
x 4
x 1
x 1
3( x 2) ( x 2)( x 2)
The LCD is 3( x 2)( x 2) .
x 1
x 1
3( x 2) ( x 2)( x 2)
3( x 2)( x 2)
6
3( x 2)( x 2)
Math 1513 Sec R.6
Simplifying Complex Rational Expressions
12
Simplify: x 3 x 3 .
34
x 1 x 2
At first, this may look like a mess. But if you break it down
and take it in steps, it is really pretty easy.
First of all, the main fraction is just a division problem, so
what we really have here is:
1 2 3 4
x 3 x 3 x 1 x 2
Each part is a simple addition or subtraction problem.
1( x 3) 2( x 3) 3( x 2) 4( x 1)
( x 3)( x 3) ( x 3)( x 3) ( x 1)( x 2) ( x 1)( x 2)
x 3 2x 6 3x 6 4x 4
( x 3)( x 3) ( x 1)( x 2)
3x 3 x 10
( x 3)( x 3) ( x 1)(x 2)
3x 3 ( x 1)( x 2)
( x 3)( x 3) x 10
(3x 3)( x 1)( x 2)
( x 3)( x 3)( x 10)
2
3( x 1) ( x 2)
( x 3)( x 3)( x 10)
7
Math 1513 Sec R.6