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Adding and Subtracting Rational Functions

What is a rational function again?

First of all, a rational function is pretty much just the division of two polynomial functions. For example, the following is a rational function:

$$ f(x)=\frac{4x+4}{6x-9} $$

How do we add or subtract them?

When adding or subtracting rational functions, you must find a common denominator as you might do with regular fractions. For example, to add \(\frac{1}{2}\) and \(\frac{1}{3}\), you might do the following:

The original fractions: $$ \frac{1}{2}+\frac{1}{3} $$

with a common denominator: $$ \frac{2}{6}+\frac{3}{6} $$

result: $$ \frac{5}{6} $$

Now, let's apply this same strategy to the addition and subtraction of rational functions:

$$ \frac{x+1}{x+3}+\frac{2x+5}{x-2} = ?? $$

Step 1) Find a common denominator by multiplying the denominators. So, \((x + 3)(x - 2)\) becomes our common denominator in this case. Then, multiply each fraction by something equivalent to "1" (which of course doesn't change the actual value!), such as \(\frac{x+3}{x+3}\), to get each fraction in terms of that common denominator:

Rational Functions

Here's what we have so far. Just multiply out (expand) the top and we will be ready to add the two fractions:

$$ \frac{(x+1)(x-2)}{(x+3)(x-2)}+\frac{(2x+5)(x+3)}{(x+3)(x-2)} $$

$$ \frac{x^2-2x+x-2}{(x+3)(x-2)}+\frac{2x^2+6x+5x+15}{(x+3)(x-2)} $$

Now add the numerators just like you would with two simple fractions:

$$ \frac{3x^2+10x+13}{(x+3)(x-2)} $$

Finally we probably want to expand the denominator as well to give us the resulting rational function:

$$ \frac{3x^2+10x+13}{x^2+x-6} $$

And that's our answer!

NOTE: To subtract rational functions, follow the same steps that you used to add rational functions, but just subtract the numerators instead of adding them!

For information on rational functions, try a search on Google or this lesson on rational functions.

By Mr. Feliz and Free Math Help

Related Pages


Rational functions may look complicated, but we can still add or subtract by finding a common denominator.