# Using Rational Expressions

A Rational Expression is the ratio of two polynomials:

## Using Rational Expressions

Using Rational Expressions is very similar to Using Rational Numbers (you may like to read that first).

## Adding Rational Expressions

The easiest way to add Rational Expressions is to use the common denominator method:

And then simplify the result.

Like in this example:

### Example:

\frac{2}{x−2} + \frac{3}{x+1} = \frac{2 × (x+1) + (x−2) × 3}{(x−2) × (x+1)}

*(Comparing to the formula above: a is 2, b is x−2, c is 3, and d is x+1)*

Then we simplify it:

= \frac{2x+2 + 3x−6}{x^{2}+x−2x−2}

= \frac{5x−4}{x^{2}−x−2}

## Subtracting Rational Expressions

Subtracting is just like Adding:

### Example:

\frac{2}{x−2} − \frac{3}{x+1} = \frac{2 × (x+1) − (x−2) × 3}{(x−2) × (x+1)}

And then simplify:

= \frac{2x+2 − (3x−6)}{x^{2}+x−2x−2}

= \frac{−x + 8}{x^{2}−x−2}

## Multiplication

To multiply two Rational Expressions, just multiply the tops and bottoms separately, like this:

### Example:

\frac{2}{x−2} × \frac{3}{x+1} = \frac{2×3}{(x−2)(x+1)}

And then simplify:

= \frac{6}{x^{2}−x−2}

## Division

To divide two Rational Expressions, first flip the second expression over (make it a reciprocal) and then do a multiply like above:

### Example:

First flip the second one over and make it a multiply:

\frac{2}{x−2} / \frac{3}{x+1} = \frac{2}{x−2} × \frac{x+1}{3}

Then do the multiply:

\frac{2}{x−2} × \frac{x+1}{3} = \frac{2(x+1)}{3(x−2)}

## Simplifying

When simplifying a rational function be careful to respect where the lower polynomial is equal to zero

### Example:

\frac{x^{2} − 1}{x + 1} is undefined when x = −1

Its Domain (the values that can go into the expression) does not include −1

Now, we can factor x^{2}−1 into (x−1)(x+1) so we get:

\frac{(x−1)(x+1)}{(x+1)}

It is now tempting to cancel (x+1) from top and bottom to produce:

x − 1

Its Domain now **does** include −1

But it is now a different function because it has a different Domain.