## Add and Subtract Rational Expressions

### Learning Outcomes

• Add and subtract rational expressions

In beginning math, students usually learn how to add and subtract whole numbers before they are taught multiplication and division. However, with fractions and rational expressions, multiplication and division are sometimes taught first because these operations are easier to perform than addition and subtraction. Addition and subtraction of rational expressions are not as easy to perform as multiplication because, as with numeric fractions, the process involves finding common denominators.

To find the least common denominator (LCD) of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, consider the following rational expressions:

$\dfrac{6}{\left(x+3\right)\left(x+4\right)},\text{ and }\frac{9x}{\left(x+4\right)\left(x+5\right)}$

The LCD would be $\left(x+3\right)\left(x+4\right)\left(x+5\right)$.

To find the LCD, we count the greatest number of times a factor appears in each denominator and include it in the LCD that many times.

For example, in $\dfrac{6}{\left(x+3\right)\left(x+4\right)}$, $\left(x+3\right)$ is represented once and  $\left(x+4\right)$ is represented once, so they both appear exactly once in the LCD.

In $\dfrac{9x}{\left(x+4\right)\left(x+5\right)}$, $\left(x+4\right)$ appears once and $\left(x+5\right)$ appears once.

We have already accounted for $\left(x+4\right)$, so the LCD just needs one factor of $\left(x+5\right)$ to be complete.

Once we find the LCD, we need to multiply each expression by the form of $1$ that will change the denominator to the LCD.

What do we mean by ” the form of $1$“?

$\frac{x+5}{x+5}=1$ so multiplying an expression by it will not change its value.

For example, we would need to multiply the expression $\dfrac{6}{\left(x+3\right)\left(x+4\right)}$ by $\frac{x+5}{x+5}$ and the expression $\frac{9x}{\left(x+4\right)\left(x+5\right)}$ by $\frac{x+3}{x+3}$.

Hopefully this process will become clear after you practice it yourself.  As you look through the examples on this page, try to identify the LCD before you look at the answers. Also, try figuring out which “form of 1” you will need to multiply each expression by so that it has the LCD.

### Example

Add the rational expressions $\frac{5}{x}+\frac{6}{y}$ and define the domain.

State the sum in simplest form.

Here is one more example of adding rational expressions, but in this case, the expressions have denominators with multi-term polynomials. First, we will factor and then find the LCD. Note that $x^2-4$ is a difference of squares and can be factored using special products.

### Example

Simplify$\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}$ and give the domain.

State the result in simplest form.

In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.

## Subtracting Rational Expressions

To subtract rational expressions, follow the same process you use to add rational expressions. You will need to be careful with signs though.

### Example

Subtract$\frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}$ and define the domain.

State the difference in simplest form.

In the next example, we will give less instruction. See if you can find the LCD yourself before you look at the answer.

### Example

Subtract the rational expressions: $\frac{6}{{x}^{2}+4x+4}-\frac{2}{{x}^{2}-4}$, and define the domain.

State the difference in simplest form.

In the previous example, the LCD was  $\left(x+2\right)^2\left(x-2\right)$.  The reason we need to include $\left(x+2\right)$ two times is because it appears two times in the expression $\frac{6}{{x}^{2}+4x+4}$.
The video that follows contains an example of subtracting rational expressions.