Using Rational Expressions
A Rational Expression is the ratio of two polynomials:
You can read more on our page on Rational Expressions
Using Rational Expressions
Using Rational Expressions is very similar to Using Rational Numbers (you might like to read that first).
Adding Rational Expressions
The easiest way to add Rational Expressions is to use the common denominator method:
 |
Like in this example:
Example:
| 2 | + | 3 | = | 2·(x+1) + (x-2)·3 |
| x-2 | x+1 | (x-2)(x+1) |
Which can then be simplified to:
| = | 2x+2 + 3x-6 | = | 5x-4 |
| x2+x-2x-2 | x2-x-2 |
Subtracting Rational Expressions
Subtracting is just like Adding:
Example:
| 2 | - | 3 | = | 2·(x+1) - (x-2)·3 |
| x-2 | x+1 | (x-2)(x+1) |
Which can then be simplified to:
| = | 2x+2 - (3x-6) | = | -x+8 |
| x2+x-2x-2 | x2-x-2 |
Multiplication
To multiply two Rational Expressions, just multiply the tops and bottoms separately, like this:
Example:
| 2 | × | 3 | = | 2·3 |
| x-2 | x+1 | (x-2)(x+1) |
Which can then be simplified to:
| = | 6 |
| x2-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:
| 2 | / | 3 | = | 2 | × | x+1 |
| x-2 | x+1 | x-2 | 3 |
Then do the multiply:
| 2 | × | x+1 | = | 2(x+1) |
| x-2 | 3 | 3(x-2) |
Simplifying
When simplifying a rational function be careful to respect where the lower polynomial is equal to zero
Example:
is undefined where 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:
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.