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Rational Expressions

I. Simplifying

II. Multiplication and Division

III. Addition

IV. Complex Fractions

Answers to Exercises

Your instructor would define a rational expression as "the quotient of two
polynomials." To you it is the source of that recurring migrane you get whenever
you work with fractions.

To be sucessful with rational expressions you must be proficient in fractional
arithmetic:

Similar problems involving rational expressions follow the same principles, only
with algebra concepts intermixed. Do you ll that headache coming on!?

I. Simplifying

Simplifying a rational expression requires finding and removing common
factors that appear in both numerator and denominator. Formally you are
using the fact that 1 ( )=( ): Informally you know it as "cancelling".
For help with factoring, see Review Topic 4.

Below are three fractions, all reduced to lowest terms.

Example:

Your own steps may differ from those shown, but there must be agreement
on the basic concept: simplifying is the removing of factors (not terms) that
are equivalent to 1.

Common Errors:

x is a term, not a factor

x² is a term, not a factor.

Repeat after me 10 times: "Factors are cancelled, not terms."

Exercise 1. Simplify each expression.

Answers

II. Multiplication and Division

There are four things to remember when multiplying or dividing fractions.

1) All fractions can be multiplied (common denominators are only neces-
sary when adding.)

2) The product of two fractions is the product of the numerators over the
product of the denominators.

3) To ensure your answer is in lowest terms, cancel common factors
before multiplying.
4) Division is defined in terms of multiplication. a

As in simplifying, most of your work comes from factoring. The rest should
be easy.

Example:

You probably were taught to cancel equal factors. Repeating the same
example:

Example: Simplify

(Remember our comment about division.)

Answer:

Exercise 2: Perform the indicated operations.

Answers

III. Addition

Illustration: Compare the following additions; notice the similarity in
steps.

(Multiplying by a form of 1 gives fractions a common denominator and still
maintains equality.)

With no common factors to cancel, answers are in lowest terms.

Common Error: Rational expressions oftentimes have polynomials for
numerators. To subtract you must \negate" each term in the numerator.

Again, we want to continue to stress that all fractions follow the same set
of rules.

Guidelines for Adding Rational Expressions

1) Express denominators in factored form.

2) Find the Least Common Denominator (LCD).

3) Convert all fractions into the common denominator by multiplying by
a form of 1.

4) Add.

5) Simplify, i.e. search for common factors to cancel.
Example: Simplify the expression

Answer:

No common factors
guarantees answer in
lowest terms.

Exercise 3: If the following represent denominators, find the LCD.

Answers

Find each sum.

Answers

IV. Complex Fractions

You are probably thinking, \Aren't all fractions complex?" Yes, but as you
will soon see, some are MORE complex.

A complex fraction contains fractions in its numerator and/or denominator.
Here are two methods you could use to simplify a complex fraction.

Example:

Method 1: Simplify any sum, then divide.

Dividing
Step

Method 2: Multiply by a form of 1 and cancel.

Exercise 4: Simplify; use both methods. Answer

Exercise 5: Simplify

Hint: Answers

Beginning of Topic 108 Skills Assessment

Simplify each expression.

Answers:

Return to Review Topic

Perform the indicated operations.

Answers:

c) Using grouping,

Return to Review Topic

If the following represent denominators, find the LCD.

a) x(x − 3) and x²(x + 3)

b) 6 − 2x and x² − 5x + 6

c) (x − 1)²; x³ − x and x²

Answers:

a) x²(x − 3)(x + 3)

b) 2(x − 3)(x − 2) Note: 3 − x = −(x − 3)

c) Since x³ − x = x(x − 1)(x + 1), LCD = x²(x − 1)²(x + 1)

( d), e), and f) are continued on next page.)

Return to Review Topic

If the following represent denominators, find the LCD.

Answers:

Return to Review Topic

Simplify; use both methods.

Answer:

Method 1:

Method 2: If we multiply by 1,

Leaving denominator in
factored form allows for
cancellation.

Return to Review Topic

Simplify

Answers:

Return to Review Topic

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