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Rational Expressions

A rational expression, also known as a rational function, is any expression or function which includes a polynomial in its numerator and denominator. In other words, a rational expression is one which contains fractions of polynomials. For example:

The last equation also has a polynomial in the denominator, keeping in mind that thus


The important thing to remember is that the denominator must never equal to zero, otherwise you'll end up dividing by zero.

When asked to find the domain of a rational function, though solving may result in many variables, you must always pick only those which will result in the polynomial in the denominator not equal to zero.

Rational Expression Examples

For example; find the domain of

What the question is asking for are the values of x for which the rational function is said to exist or make mathematical sense. In other words, find the values of x for which the denominator is not equal to zero. So the first step is equating the denominator to zero i.e.

from which you can see that

and then we say that the domain is: all values of x except for x = 3

Notice on the graph of the function, we have an asymptote at x = 3 which means that this value is not in the domain. If it is not in the domain, then a range value (y-value) cannot exist.

Example: Find the domain of the expression below

As before, start with equating the denominator to zero and then find factor the resulting equation to find its roots

which means that the roots of the denominator are

These are the values for which the denominator is equal to zero, thus we say that the domain of the expression is given by:

all values of x except

Example: Find the domain of

Equate the denominator and factor

so the whole rational expression becomes

Although we have expressions in both the denominator and denominator, the expression in the numerator does not affect the domain of the entire rational expression, so we only consider the denominator


which means that x = {1,3,4}

And thus the domain of the rational expression is:

all values of x except for x = {1,3,4}

Simplifying Rational Expressions

Rational Expressions can be factored and simplified as in the example below:

First factor both numerator and denominator

then you can see that x is a common factor in both the numerator and denominator, so the above is the same as:

However, it is important to remember you should never simplify the rational expression before finding the domain. In case you still feel like simplifying before finding the domain, then you must keep track of the factors which you 'cancel' out.

For the example above, to find the domain from the simplified expression

set the denominator equal to zero, then solve for x

from which

However, x = 2/3 is not the only factor for which the denominator of 3x/(2x - 3x2) is equal to zero. Since we divided through by a factor to get the simplified expression, we must set that factor to zero as well and solve for x.

In this case since we divided through by x, we say

and then we give the domain as: all values of x except for x = {0,2/3}

Example: Simplify the rational expression and the also state the domain

Step 1

First factor both the numerator and denominator to get

Step 2

In this form, it should be easy to see the common factors

Step 3

but (x - 3) and (3 - x) are very similar can can be manipulated so that we can also cancel them from the expression

Step 4

factoring out -1 gives

Step 5

substituting the above into the expression

the above is the simplified expression needed.

Step 6

To find the domain; equate the factors to zero to get the points where the denominator will be zero i.e.

and the domain is given as:

all values of x except


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