The following were originally developed by Margaret Morgan and are being used with her permission.
 
 

Lecture : Rational Expressions

What is a rational expression?

Note that the word rational contains the word ratio. We associate the word ratio with fractions. So, a rational expression is a fraction in which both the numerator and the denominator are polynomials. Do you recall what a polynomial is? Write down an example.

In addition to rational expressions, in future chapters we will study rational equations. It is important to note the difference between a rational expression and a rational equation, because we work with them differently. A rational equation has an = symbol, in it we try to solve for a variable, a rational expression does not have an equal sign, with it we simplify perform the indicated operation, addition, subtraction, multiplication or division.

Here is an example of a rational expression:

 

 

 

Determine whether each of the following is a rational expression or a rational equation.

 

a)

 

 

b)

c)

 

d)

e)

 

 

 

Solutions:

a. Rational expression

b. Rational equation

c. Rational expression

d. Rational equation

e. Rational expression
 

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Lecture : Simplifying Rational Expressions

In order to simplify a radical expression, we factor both the numerator and the denominator and then cancel any like terms.

Example:

First let us factor the numerator, note that it is a difference of cubes.

Now we factor the denominator, it has a common factor of 2 which we must factor out first.

So, we have a difference of squares to factor.

The numerator and denominator are both completely factored. The only factor in common to the numerator and the denominator is x-5, we will cancel it from both to get our answer.

Simplify each of the following.

1.

: solution

2.

:solution

3.

: solution
 

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#1 solution

1.

First we factor a 3 out of the numerator

To finish factoring the numerator we may use trial and error or the ac method.

To factor the denominator, we need only factor out the common factor of 9.
 

Now that we have finished factoring, we cancel like terms. We can cancel a 3, leaving us with a 1 we do not need to write in the numerator, and a 3 in the denominator. We may also cancel (x - 3) from both.


 
 

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#2 solution

2.

Here we have a difference of cubes in the numerator and a difference of squares in the denominator.

And we may cancel the 3x-2 to get our answer.


 
 

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#3 solution

3.

In the numerator we have 4 terms which indicates that we must use factoring by grouping.

To factor the denominator we need factors of 6 which add to be 5. 2 and 3 will work.

Now we cancel both (x+2) and (x + 3) and are left with x-2 as our answer.
 

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