Rational Expressions
A rational expression is a quotient of two polynomials P and Q, with . A rational expression is a ratio of two polynomials in which the denominator is not the zero polynomial. The denominator must also have a variable in it to be considered a rational expression. The domain of a rational expression includes any number that DOES NOT make the denominator zero. (Remember: you cannot divide by zero.)
Example:
Objectives:
-Find the domain of a rational expression
-Putting an expression in lowest terms
-Multiplying and dividing rational expressions
-Adding and subtracting rational expressions
-Simplifying complex fractions

The domain of a rational expression can be found by setting the factored denominator equal to zero and solving the resulting equation.

For the example above, you would set x + 2 = 0 and get x = 2. Therefore, the domain is every number except x = 2. You can write the domain in two ways…
Set notation: This reads…x such that x cannot equal 2.
Interval notation:

Example 1: Find the domain of each rational expression.
(a)
Since the denominator is already factored, we can set each part equal to zero and find our exclusions for the domain. This expression can actually be simplified as well, but you ALWAYS find the domain before you simplify the expression. and
Domain: or
(b)
First, factor the denominator. factors into

Now set both parts equal to zero and
Domain:

***Most of the time, set notation is used for domain. Just be sure to be familiar with both types of notations!!!

Example 2: Write each expression in lowest terms. Remember, lowest terms means to cancel out any common factors from the numerator and denominator.
(a)
First, we must factor the numerator and denominator.

Then cancel out any common factors. Here, x + 4 is a common factor.
Answer:
**Common mistake is that students try to cancel out the x’s in the solution. You cannot do this because the x in the numerator is part of a binomial expression. So if you have any addition or subtraction signs, you must be able to cancel out the entire polynomial not just part of it.

(b)
Factor by pulling out a GCF in the numerator and using the difference of squares in the denominator.

This is a special case. You can see how you would want to cancel out the 2 – x in the numerator with the x – 2 in the denominator but these are not exactly the same. What can you do to fix it? Factor out a negative one which will change the signs of what is in the parentheses.

Now we can cancel!
Answer:

Example 3: More reducing…
(be sure to find domain before you simplify the expression if they ask for it)
(a) (b) (c)
Step 1: Factor both numerator and denominator if possible.
Step 2: Cancel out common factors (remember: a factor is a term of multiplication.)

(a) (b) (c)

Multiplying and dividing rational expressions: It helps to reduce each expression to lowest terms before multiplying or dividing. It keeps the numbers smaller.

Example 4: Multiply each expression (a) (b)
Step 1: Factor as much as possible
Step 2: Cancel out common factors either by a diagonal or top to bottom
Step 3: Multiply out what is left
(a) (b)

Example 5: Divide each expression
(a) (b)
Step 1: Keep first fraction the same, change the division sign to multiplication, and then flip the second fraction.
Step 2: Continue to factor and reduce like you did in example 3
(a) (b)

Example 6: Multiply or divide using the above methods…
(a)

Answer:

(b)
This problem is much more complicated in terms of the factoring…here you need the difference of cubes formula, the difference of squares formula and factor by grouping. Please review section R.4 if you need any help in getting these factored.

Answer:

Adding and Subtracting rational expressions: you must “build up the denominator” in order to add or subtract. The denominators must be the same. (Get the LCD). Remember, to get the LCD, multiply both the numerator and the denominator by the same nonzero number to get equivalent fractions.

Procedure for finding the LCD:
Factor each denominator completely.
Write a product using each factor that appears in the denominator.
For each factor, use the highest power of that factor that occurs in the denominators. (Do not add them together.)

Example 7: Add each expression.
(a) (b) (c)
Step 1: Find the LCD
Step 2: Figure out what you need to make each term the same and multiply by that expression
Step 3: Add/Subtract the numerators and put over the LCD
(a) (b) (c)

Example 8: Subtract each expression.

(a)

(b)

Complex Fractions are fractions that have a rational expression in both the numerator and the denominator. When simplifying complex fractions, it is best to treat the numerator and the denominator as separate problems, simplify individually, and then combine them together.

Example 9:
Step 1: Write both the numerator and denominator as a single fraction (if you have more than one term).
Step 2: Rewrite the first fraction (top) and then change to multiplication and flip the second fraction (bottom).
Step 3: Factor and cancel.
(a)

(b)

(c)