# 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:

2x−2 + 3x+1  =  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:

=   2x+2 + 3x−6 x2+x−2x−2

=   5x−4 x2−x−2

## Subtracting Rational Expressions

Subtracting is just like Adding:

### Example:

2 x−2 3 x+1   =   2 × (x+1) − (x−2) × 3 (x−2) × (x+1)

And then simplify:

=   2x+2 − (3x−6) x2+x−2x−2

=   −x + 8 x2−x−2

## Multiplication

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

### Example:

2x−2  ×  3x+1  =  2×3(x−2)(x+1)

And then simplify:

=  6x2−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:

2x2   /   3x+1  =  2x2  ×  x+13

Then do the multiply:

2 x2   ×  x+13  =  2(x+1)3(x2)

## Simplifying

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

### Example:

x2 − 1x + 1 is undefined when x = −1

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

Now, we can factor x2−1 into (x−1)(x+1) so we get:

(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.

474, 475, 476, 477, 2288, 2289, 2290, 2291, 1120, 1121