Adding and subtracting rational expressions brings everything you learned about fractions into the world of algebra. We will mix common denominators with factoring and FOILing.

## Rational Polynomial Defined

The word ‘rational’ means ‘fraction.’ So a rational polynomial is a fraction with polynomials in the numerator (top) and/or denominator (bottom). Here’s an example of a rational polynomial:

(*x* + 4) / (*x*^2 + 3*x* + 2)

As we get started, let’s remember that to add or subtract fractions, we need a common denominator. Try this mnemonic to help you remember when you need a common denominator and when you don’t:

*Add Subtract Common Denominators; Multiply Divide None*

*Auntie sits counting diamonds; Mother does not.*

Let’s get started!

## Adding and Subtracting Rational Expressions

- We need to factor.
- Find a common denominator.
- Rewrite each fraction using the common denominator.
- Put the entire numerator over the common denominator.
- Simplify the numerator.
- Factor and cancel if possible.
- Write the final answer in simplified form.

There are quite a few steps, but let me show you how they work.

## Example #1

Our first expression is (1 / (*x* – 2)) + (3 / (*x* + 4)).

The first step is to factor. Since we don’t have anything to factor, let’s move to the next step, writing down our denominators, (*x*-2) and (*x*+4). This will be our common denominator: (*x* – 2)(*x* + 4).

Now we need to create our common denominator. Let’s look at our first term, (1 / (*x* – 2)). (*x* – 2) is in the denominator. We need to multiply by (*x* + 4) to make our common denominator. But if we multiply by (*x* + 4) on the bottom, we need to multiply by (*x* + 4) on the top.

For right now we are going to write it and not multiply yet.

Let’s look at our second term: (3 / (*x* + 4)). The denominator is (*x* + 4). We need to multiply (*x* – 2) times (*x* + 4) to get our common denominator. But once again, if we multiply by (*x* – 2) on the bottom, we need to multiply by it on the top too.

So far, this is what we have:

((1(*x* + 4)) / ((*x* – 2)(*x* + 4))) + ((3(*x* – 2)) / ((*x* + 4)(*x* – 2)))

Don’t FOIL the denominator – we may have to cancel as our final answer!

Now let’s write the entire numerator over our common denominator.

(1(*x* + 4)) + 3(*x* – 2)) / ((*x* – 2)(*x* + 4))

Let’s simplify the numerator.

1(*x* + 4) = *x* + 4

3(*x* – 2) = 3*x* – 6

(*x* + 4 + 3*x* – 6) / ((*x* + 4)(*x* – 2))

Collect like terms in the numerator.

(4*x* – 2) / ((*x* + 4)(*x* – 2))

Factor the numerator if possible.

4*x* – 2 = 2 (2*x* – 1)

(2(2*x* – 1)) / ((*x* + 4)(*x* – 2))

There isn’t anything to slash or cancel, so we distribute and FOIL for our final answer.

(4*x* – 2) / (*x*^2 + 2*x* – 8)

## Example #2

((2*x*) / (*x*^2 – 16)) – (1 / (*x* + 4))

*x*^2 – 16 factors into (*x* – 4)(*x* + 4). So let’s put that into the expression.

((2*x*) / ((*x* – 4)(*x* + 4))) – (1 /(*x* + 4))

Our next step is to write down all of our denominators.

In the first term, we have (*x* + 4)(*x* – 4), so we write those down.

We continue to the next term and look at the denominator. We never duplicate denominators from term to term. Since we already have (*x* + 4) written as part of our denominator, we don’t need to duplicate it. So it turns out our common denominator will be (*x* + 4)(*x* – 4).

Now we need to create our common denominator. Let’s look at our first term ((2*x*) / (*x* + 4)(*x* – 4)). We already have our common denominator here, so we’re going to move to the next term: (1 / (*x* + 4)).

Here, we need to multiply (*x* – 4) to make our common denominator. But if we multiply (*x* – 4) on the bottom, we need to multiply by (*x* – 4) on the top. For right now, we are going to write it and not multiply yet. So we have ((2*x*) / (*x* + 4)(*x* – 4)) – (1(*x* – 4) / (*x* + 4)(*x* – 4)).

Let’s write the numerator all over the denominator.

((2*x*)-1(*x*-4))/((*x*+ 4)(*x* – 4))

Simplify the numerator (or top) and rewrite it over the denominator.

Distribute the -1 into (*x* – 4) = -1*x* + 4.

Collecting like terms, 2*x* – 1*x*= *x*.

So now our expression looks like:

(*x* + 4) / (*x*+ 4)(*x*– 4)

We can slash, or cancel, (*x*+ 4) over (*x*+ 4).

This gives us 1/(*x* – 4) as our final answer.

## Example #3

((5*x*^2 – 3) / (*x*^2 + 6*x* + 8)) – 4

The first step is to factor.

*x*^2 + 6*x* + 8 = (*x* + 4)(*x* + 2)

Our next step is to write down all of our denominators.

In our first term, we have (*x* + 4)(*x* + 2), so we write it down.

The denominator for the next term is 1.

Therefore, our common denominator will be (*x* + 4)(*x* + 2).

Now we need to create our common denominator. Let’s look at our first term (5*x*^2 – 3)/((*x* + 4)(*x* + 2)).

We already have our common denominator here, so we’re going to move to the next term, 4.

Here, we only have a 1 in the denominator, so we need to multiply by (*x* + 4)(*x* + 2) over (*x*+4)(*x*+2).

This is what our new expression is going to look like:

((5*x*^2 – 3) / (*x* + 4)(*x* + 2)) – ((4 (*x* + 4)(*x* + 2)) / ((*x* + 4)(*x* + 2))) .

Let’s write the whole numerator (top) over the denominator (bottom).

((5*x*^2 – 3 – 4(*x* + 4)(*x* + 2))) / ((*x* + 4)(*x* + 2))

We can now simplify the top, or numerator.

(*x*+4)(*x*+2) = *x*^2 +6*x* +8

Multiply -4( *x*^2 +6*x* +8) and we have -4*x*^2 – 24*x* – 32.

Let’s continue with the numerator and collect like terms, so our expression looks like:

(*x*^2 – 24*x* – 35) / ((*x* + 4)(*x* + 2))

The numerator does not factor without using the quadratic formula, so this is almost our answer, except we need to FOIL the bottom, or denominator. Here is our final answer:

(*x*^2 – 24*x* – 35) / (*x*^2 + 6*x* + 8)

## Lesson Summary

As we have seen, the process to add or subtract rational expressions is:

- We need to factor.
- Find a common denominator.
- Rewrite each fraction using the common denominator.
- Put the entire numerator over the common denominator.
- Simplify the numerator.
- Factor and cancel if possible.
- Write the final answer in simplified form.

## Lesson Objectives

Once you complete this lesson you’ll be able to add or subtract rational expressions.