# Adding & subtracting rational expressionsÂ (advanced)

CCSS Math: HSA.APR.D.7

Have you learned the basics of rational expression addition/subtraction? Great! Now dig deeper with some advanced examples.

#### What we need to know before this lesson

A

**rational expression**is a ratio of two polynomials.To add or subtract two rational expressions with the

*same*denominator, we simply add or subtract the numerators and write the result over the common denominator.When the denominators are

*not*the same, we must manipulate them so that they*become*the same. In other words, we must find a common denominator.If this is new to you, you may want to check out the following articles first:

#### What you will learn in this lesson

In this lesson, you will practice adding and subtracting rational expressions with different denominators. You will use the

**least common denominator**as your common denominator in these examples and explore why it is beneficial to do so.## Warm-up: $\dfrac{3}{x-2}-\dfrac{2}{x+1}$

To subtract two rational expression, each fraction must have the same denominator!

In this example, we can create a common denominator by multiplying the first fraction by $\left({\dfrac{x+1}{x+1}}\right)$ and the second fraction by $\left({\dfrac{x-2}{x-2}}\right)$.

Then, we can subtract the numerators and write the result over the common denominator.

### Check your understanding

## Least common denominators

### Numerical fractions

Sometimes, the denominators of two fractions are different but have some shared factors.

For example, consider $\dfrac{3}{4}+\dfrac{1}{6}$:

Notice that the common denominator used in this example was

*not*the product of the two individual denominators $(24)$. Instead it was the**least common multiple**of $4$ and $6$ $(12)$.The least common multiple of the denominators in two or more fractions is called the

**least common denominator**.### Variable expressions

Now let's apply this reasoning to perform the following addition:

First, let's find the least common denominator:

So the least common denominator is $\blueD{(x-2)}\greenD{(x+1)}\purpleC{(x+3)}$.

We can add the rational expressions as follows:

### Check your understanding

### Challenge Problem

## Why use the least common denominator?

You may be wondering why it is so important to use the

*least common denominator*to add or subtract rational expressions.After all, this is not a requirement, and it is easy enough to use other denominators with numerical fractions.

For example, the table below calculates $\dfrac{3}{4}+\dfrac{1}{6}$ using two different common denominators â€” one using the

*least common denominator*$(12)$ and the other using the product of the two denominators $(24)$.Least common denominator $(12)$ | $~~~~~$Common denominator $(24)$ |
---|---|

$\begin{aligned}~\dfrac{3}{4}+\dfrac{1}{6}&=\dfrac{3}{\blueD4} {\left(\dfrac{\goldD3}{\goldD3}\right)}+\dfrac{1}{\greenD6}{\left(\dfrac{\purpleD2}{\purpleD2}\right)}\\\\&=\dfrac{9}{12}+\dfrac{2}{12}\\\\&=\dfrac{11}{12}\\\\&\phantom{\dfrac{1}{2}}\end{aligned}$ | $~~~~~~~\begin{aligned}\dfrac{3}{\blueD4}+\dfrac{1}{\greenD6}&=\dfrac{3}{\blueD4}\left(\greenD{\dfrac{6}{6}}\right)+\dfrac{1}{\greenD6}\left(\blueD{\dfrac{4}{4}}\right)\\\\&=\dfrac{18}{24}+\dfrac{4}{24}\\\\&=\dfrac{22}{24}\\\\&=\dfrac{11}{12}\end{aligned}$ |

Notice that when using $24$ as the common denominator, more work was required. The numbers were larger and the resulting fraction needed to be simplified.

This will also happen if you do not use the least common denominator when adding or subtracting rational expressions.

However, with rational expressions, this process is much more difficult because the numerators and denominators will be polynomials instead of integers! You will have to perform arithmetic with higher degree polynomials and factor polynomials to simplify the fraction.

All of this extra work can be avoided by using the

*least common denominator*when adding or subtracting rational expressions.