Rational Expressions

Sometimes an expression comes in the form of a fraction. For example:

This algebraic expression is a rational expression. Rational expressions can sometimes be simplified.

When an expression comes in this form, common factors in the numerator and denominator cancel out just like with regular fractions.

When one part of the fraction comes in expanded form, the expanded part has to be put in factored form in order to cancel out the factors.

Numerator:

x²+(2+1)x+2*1
x²+2x+x+2
(x+2)x+(x+2)
(x+2)(x+1)
x²+3x+2 = (x+2)(x+1)

The rational expression in factored form is:

Notice that the factors in the numerator match the factors in the denominator. In this case, all the factors cancel out.

Sometimes the rational expression is in expanded form in both the numerator and the denominator.

When it is factored it becomes:

where

The result is

Multiplying rational expressions is just like multiplying regular fractions. The numerator of one fraction multiplies with the numerator of the other. The denominator of one fraction multiplies with the denominator of the other.

Rational expressions can also be added:

To add this expression, the denominators have to be the same. To do this, the first thing to do is to multiply each term by a fraction equal to one.

The fractions of ones now have to be set to the factors in the denominators of every other term. So:

becomes

The final form would be

Now that the denominators are the same, the fractions can be added.

The like terms are added together for the final answer.