# Simplifying Rational Expressions

A rational expression is an algebraic expression can be written as the quotient of two polynomials (Hence any polynomial is also a rational expression). Rational expression domain is set of all real values of the variables except those that make the denominator equal to zero.

In this lesson, you'll learn how to simplify rational expressions using your skills of finding the greatest common factor and factoring the polynomials out this common factor.

## Lowest Terms of a Rational Expression

A rational expression is written in lowest terms if the greatest common factor of its numerator and denominator is 1 or -1. To write a rational expression we must use the Fondamental Principle of Fractions, then reducing the rational expression to the lowest terms by dividing out the greatest common factor.

### Note:

Fondamental Principle of Fractions:

Let are real numbers with

## Procedure to Simplify Rational Expressions

A rational expression is simplified if it is reduced to the lowest terms (Its denominator and numerator have no common factor other than 1 or -1).

### Note:

Simplify Rational Expressions:

1. Factor the numerator and the denominator completely using the common factor.

2. Divide both the numerator and the denominator out the common factor.

## Examples of how to Simplify Rational Expressions

This is a list of examples of how to reduce rational expressions to a lowest terms. The skill required in these examples is to be able to find the greatest common factor of the numerator and denominator.

#### Example1:

Simplify the following rational expression:

#### Solution:

Factoring the denominator: The expression at denominator is a Factoring difference of two perfect squares, so:

Factoring the numerator: the expression at the numerator is a polynomial can be factored as product of two binomials:

Note carefully that there is a common factor between the two obtained expressions which is .

Hence,

#### Example2:

Simplify the following rational expression:

#### Solution:

Factoring the denominator: The expression at denominator can be written as a difference of two perfect squares:

Factoring the numerator: Factoring the numerator out the greatest common factor between the two terms and which is 3.

Now, I assume that you noted the common factor wich is

Hence,

#### Example3:

Simplify the following rational expression:

#### Solution:

Factoring the denominator: The numerator can be factored using the greatest common factor of and which is .

Factoring the numerator: The numerator is a trinomial expression, we can factor it using quadratic formula. So, we must calculate the descriminant:

The quadratic expression has two roots:

Therefor;

Note carefully that there is a common factor between the two obtained expressions which is .

Hence;

#### Example4:

Simplify the following rational expression:

#### Solution:

Factoring the denominator: The numerator is a difference of two perfect squares, so it can be factored as follow:

Note that there is already a common factor between the denominator and the numerator which is .

#### Example4:

Simplify the following rational expression:

#### Solution:

Factoring the denominator: The numerator is a difference of two perfect squares, so it can be factored as follow:

Factoring the numerator: The numerator is a binomial expression can be factored as follow:

But for a moment, the denominator and the numerator don't have a common factor, right? If you notice carefully when we factored by the number 3 we could also have factored with the number -3 because in this case, we will get a common factor.

Now, I assume that you noted the common factor wich is . Hence,

#### Example5:

Simplify the following rational expression:

#### Solution:

Factoring the numerator: No change will occur to the numerator because it is not factorizable as a product of two binomials.

Factoring the denominator: Notice that the denominator is a trinomial which the three terms have a common factor which is .

But up to here, we still don’t get a common factor of the numerator and denominator. If you notice well that the expression is a quadratic expression that can be factored as follows:

So, the denominator can factorized as follow:

Finally, the rational expression can be simplified as follow: