Adding and Subtracting Rational Expressions, in Depth

In order to be able to add and subtract rational expressions, it's important that you first understand what rational expressions are. Our next step is being able to combine them!

Combining Fractions with Different Denominators

Before we dive into adding and subtracting rational expressions, let's go back to adding and subtracting fractions with unlike denominators. Remember how to combine fractions with like denominators, where we added the numerators together over the same denominator. With unlike denominators, we had to first find the Least Common Multiple in order to have the same denominator.

If we want to find the sum of $\frac34 + \frac12$, we need to find a common denominator. We can see that we can combine the second fraction by $\frac22$.

$$\frac34 + \frac12 = \frac34 + \frac24 = \frac 54$$

Extending to Rational Expressions

We use this same concept when we are evaluating rational expressions! Since rational expressions are essentially a ratio of two expressions, we set up the expressions like a fraction.

$$\frac{1}{3x+2}$$

Let's think about what would happen if we wanted to add two rational expressions together. Then we're adding two fractions together!

$$\frac{1}{3x+2} + \frac{4}{2x-1}$$

Right now, we can't add the two rational expressions. That’s because they do not have the same denominator.

To add or subtract rational expressions, the denominators must be the same. This rule is just like adding regular fractions.

Let's think about what we can do.

We need to find a Least Common Multiple (LCM) of the denominators. The LCM helps us create a common denominator so we can combine the expressions.

Once the denominators match, we can add or subtract the numerators.

Let's start with the first fraction, $\frac{1}{3x+2}$. We can multiply the numerator and denominator both by $2x-1$.

$$\begin{align} \frac{1}{3x+2} \cdot \frac{2x-1}{2x-1} \\ \frac{1 \cdot (2x-1)}{(3x+2) \cdot (2x-1)} \\ \frac {2x-1}{6x^2-3x+4x-2} \\ \frac {2x-1}{6x^2+x-2} \\ \end{align}$$

We're now going to do the same thing for the second fraction!

$$\begin{align} \frac{4}{2x-1} \cdot \frac{3x+2}{3x+2} \\ \frac{4 \cdot (3x+2)}{(2x-1)(3x+2)} \\ \frac{12x+8}{6x^2-3x+4x-2} \\ \frac{12x+8}{6x^2+x-2} \\ \end{align}$$

Now that we've got two fractions with the same denominator, we can add them together!

$$\frac {2x-1}{6x^2+x-2} + \frac{12x+8}{6x^2+x-2}$$

This new expression probably looks very daunting, but remember that you did the bulk of the work by ensuring both fractions have the same denominator. Now you can add the numerators!

$$\begin{align} \frac {(2x-1)+(12x+8)}{6x^2+x-2} \\ \frac {2x-1 +12x+8}{6x^2+x-2} \\ \frac {14x+7}{6x^2+x-2} \\ \end{align}$$

We solved it!

$$\frac{1}{3x+2} + \frac{4}{2x-1} = \frac {14x+7}{6x^2+x-2}$$

We use the same process when subtracting rational expressions.

First, make sure the denominators are the same. This is called finding the Least Common Denominator (LCD).

Once the denominators match, you simply subtract the numerators. The denominator stays the same.

You do not need to learn any new rules. Adding and subtracting rational expressions with a common denominator works just like regular fractions.

(Video) Adding and Subtracting Rational Expressions

by patrickJMT

In this video by patrickJMT, you can walk through an example of how to add rational expressions that have unlike denominators.

Summary

Example: Evaluate the following rational expression: $\frac{7}{x^2-64} + \frac{3}{x+8}$

Our first step is to find a common denominator. To make that easier, we will make sure that all of our denominators are in factored form.

$$\frac{7}{(x+8)(x-8)} + \frac{3}{x+8}$$

We now want to think of the least common multiple of the denominators. If we multiply the second fraction by $\frac{x-8}{x-8}$, the denominators will be the same.

$$\frac{7}{(x+8)(x-8)} + \frac{3}{x+8} \cdot \frac{x-8}{x-8}$$

We now have a common denominator! We can now add the numerators of the fraction.

$$\frac{7+3(x-8)}{x^2-64}$$

This fraction simplifies to:

$$\frac{7+3x-24)}{x^2-64}$$

We can simplify one more time:

$$\frac{3x-17)}{x^2-64}$$

We have successfully combined both fractions in the original rational expression into one fraction!

Adding and Subtracting Rational Expressions

Next up in our lessons on evaluating rational functions, we will learn how to add and subtract rational expressions. A rational expression is a fraction with polynomials.

Just like with regular fractions, the denominators must be the same before you add or subtract.

If the denominators are already the same, you can simply add or subtract the numerators. The denominator stays the same.

If the denominators are different, you must first find a common denominator. Once they match, you can combine the expressions.

In the above example, we want to add $\frac{1}{x} + \frac{1}{x+1}$. The common denominator would be $x(x+1)$. We rewrite both fractions to have that bottom, and then we can add the top parts together.

Always simplify your answer at the end as much as you can.

Looking Further

In the next few lessons, you will learn to multiply and divide rational expressions and simplifying complex rational expressions. We will then dive deeper into solving rational equations.