How to Add And Subtract Rational Expressions

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 + 3x + 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) = 3x - 6 (x + 4 + 3x - 6) / ((x + 4)(x - 2))

Collect like terms in the numerator.

kkk

(4x - 2) / ((x + 4)(x - 2)) Factor the numerator if possible.

4x - 2 = 2 (2x - 1) (2(2x - 1)) / ((x + 4)(x - 2))

There isn't anything to slash or cancel, so we distribute and FOIL for our final answer. (4x - 2) / (x^2 + 2x - 8)

Example #2

((2x) / (x^2 - 16)) - (1 / (x + 4)) x^2 - 16 factors into (x - 4)(x + 4). So let's put that into the expression.

((2x) / ((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 ((2x) / (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 ((2x) / (x + 4)(x - 4)) - (1(x - 4) / (x + 4)(x - 4)).

Let's write the numerator all over the denominator. ((2x)-1(x-4))/((x+ 4)(x - 4))

Simplify the numerator (or top) and rewrite it over the denominator. Distribute the -1 into (x - 4) = -1x + 4. Collecting like terms, 2x - 1x= 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

((5x^2 - 3) / (x^2 + 6x + 8)) - 4 The first step is to factor.

x^2 + 6x + 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 (5x^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: ((5x^2 - 3) / (x + 4)(x + 2)) - ((4 (x + 4)(x + 2)) / ((x + 4)(x + 2))) .

kkk

Let's write the whole numerator (top) over the denominator (bottom). ((5x^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 +6x +8 Multiply -4( x^2 +6x +8) and we have -4x^2 - 24x - 32.

Let's continue with the numerator and collect like terms, so our expression looks like: (x^2 - 24x - 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 - 24x - 35) / (x^2 + 6x + 8)