What Are Rational Algebraic Expressions?

Rational Algebraic Expression

Any algebraic expression, that is a quotient of two other algebraic expressions, is called a rational algebraic expression. Thus if P and Q are algebraic expressions and Q ¹ 0 then the expression     is called a rational algebraic expression.

We say that a rational algebraic expression  is meaningless for those values of the variable for which the denominator Q is zero.

To simplify rational algebraic expressions, we may be required to factorize algebraic expressions in the numerator and /or denominator.

Operations with algebraic fractions (i.e. rational algebraic expressions)

Simplifying RAE

Thinking back to when you were dealing with whole-number fractions, one of the first things you did was simplify them: You "cancelled off" factors which were in common between the numerator and denominator. You could do this because dividing any number by itself gives you just "1", and you can ignore factors of "1".

Using the same reasoning and methods, let's simplify some rational expressions.

* Simplify the following expression:

    To simplify a numerical fraction, I would cancel off any common numerical factors. For this rational expression (this polynomial fraction), I can similarly cancel off any common numerical or variable factors.

The numerator factors as (2)(x); the denominator factors as (x)(x). Anything divided by itself is just "1", so I can cross out any factors common to both the numerator and the denominator. Considering the factors in this particular fraction, I get:

Then the simplified form of the expression is: 

* Simplify the following rational expression: 

How nice! This one is already factored for me! However (warning!), you will usually need to do the factorization yourself, so make sure you are comfortable with the process!

The only common factor here is "x + 3", so I'll cancel that off and get:

Then the simplified form is:

Warning: The common temptation at this point is to try to...