## Adding and Subtracting Rational Expressions Part I

### Learning Outcomes

• Add and subtract rational expressions with like denominators
• Add and subtract rational expressions with unlike denominators using a greatest common denominator

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. By working carefully and writing down the steps along the way, you can keep track of all of the numbers and variables and perform the operations accurately.

## Adding and Subtracting Rational Expressions with Like Denominators

Adding rational expressions with the same denominator is the simplest place to start, so let’s begin there.

To add fractions with like denominators, add the numerators and keep the same denominator. Then simplify the sum. You know how to do this with numeric fractions.

$\begin{array}{c}\frac{2}{9}+\frac{4}{9}=\frac{6}{9}\\\\\frac{6}{9}=\frac{3\cdot 2}{3\cdot 3}=\frac{3}{3}\cdot \frac{2}{3}=1\cdot \frac{2}{3}=\frac{2}{3}\end{array}$

Follow the same process to add rational expressions with like denominators. Let’s try one.

### Example

Add $\displaystyle \frac{2{{x}^{2}}}{x+4}+\frac{8x}{x+4}$, and define the domain.

State the sum in simplest form.

Caution!  Remember to define the domain of a sum or difference before simplifying.  You may lose important information when you simplify. In the example above, the domain is $x\ne-4$.  If we were to have defined the domain after simplifying, we would find that the domain is all real numbers which is incorrect.

To subtract rational expressions with like denominators, follow the same process you use to subtract fractions with like denominators. The process is just like the addition of rational expressions, except that you subtract instead of add.

### Example

Subtract$\frac{4x+7}{x+6}-\frac{2x+8}{x+6}$, and define the domain.

State the difference in simplest form.

### Try It

In the video that follows, we present more examples of adding rational expressions with like denominators. Additionally, we review finding the domain of a rational expression.

## Adding and Subtracting Rational Expressions with Unlike Denominators

What do they have in common?

Before adding and subtracting rational expressions with unlike denominators, you need to find a common denominator. Once again, this process is similar to the one used for adding and subtracting numeric fractions with unlike denominators. Remember how to do this?

$\displaystyle \frac{5}{6}+\frac{8}{10}+\frac{3}{4}$

Since the denominators are $6$, $10$, and $4$, you want to find the least common denominator and express each fraction with this denominator before adding. (BTW, you can add fractions by finding any common denominator; it does not have to be the least. You focus on using the least because then there is less simplifying to do. But either way works.)

Finding the least common denominator is the same as finding the least common multiple of  $4$, $6$, and $10$. There are a couple of ways to do this. The first is to list the multiples of each number and determine which multiples they have in common. The least of these numbers will be the least common denominator.

Number

Multiples

$4$ $8$ $12$ $16$ $20$ $24$ $28$ $32$ $36$ $40$ $44$ $48$ $52$ $56$ $\textbf{60}$ $64$
$6$ $12$ $18$ $24$ $30$ $36$ $42$ $48$ $54$ $\textbf{60}$ $66$ $68$
$10$ $20$ $30$ $40$ $50$ $\textbf{60}$  $70$ $80$

The other method is to use prime factorization, the process of finding the prime factors of a number. This is how the method works with numbers.

### Example

Use prime factorization to find the least common multiple of $6$, $10$, and $4$.

Both methods give the same result, but prime factorization is faster. Your choice!

Now that you have found the least common multiple, you can use that number as the least common denominator of the fractions. Multiply each fraction by the fractional form of $1$ that will produce a denominator of $60$:

$\begin{array}{r}\frac{5}{6}\cdot \frac{10}{10}=\frac{50}{60}\\\\\frac{8}{10}\cdot\frac{6}{6}=\frac{48}{60}\\\\\frac{3}{4}\cdot\frac{15}{15}=\frac{45}{60}\end{array}$

Now that you have like denominators, add the fractions:

$\frac{50}{60}+\frac{48}{60}+\frac{45}{60}=\frac{143}{60}$

In the next example, we show how to find the least common multiple of a rational expression with a monomial in the denominator.

### Example

Add$\frac{2n}{15m^{2}}+\frac{3n}{21m}$, and give the domain.

State the sum in simplest form.

That took a while, but you got through it. Adding rational expressions can be a lengthy process, but taken one step at a time, it can be done.  Now let’s take a look at some examples where the denominator is not a monomial.

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 where the denominators are 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.

The video that follows contains an example of adding rational expressions whose denominators are not alike.  The denominators of both expressions contain only monomials.

## Subtracting Rational Expressions

Now let’s try subtracting rational expressions. You’ll use the same basic technique of finding the least common denominator and rewriting each rational expression to have that denominator.

### Example

Subtract$\frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}$, 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 whose denominators are not alike.  The denominators are a trinomial and a binomial.

### Try it

On the next page, we will show you how to find the greatest common denominator for a rational sum or difference that does not share any common factors.  We will also show you how to manage a sum or difference of more than two rational expressions.