It is my humble opinion that one of the "best"
parts of K. Elayn Martin-Gay's presentation of algebra (Introductory
Algebra) is how she prepares students for factoring polynomials
(Chapter 4), moves students through that section, and then immediately
uses those concepts to introduce rational expressions (Chapter 5).
In many ways, her presentation of rational
expressions if a complete review of Chapter 4. It is my experience
that Chapter 4 and 5 are the key to success in Beginning Algebra.
I believe that a full understanding of these sections also determines
who is fully ready for Intermediate Algebra 141.
As we work through this chapter, please stop
periodically and ask yourself, "what parts of this chapter
represented a review and what part was really 'new'." Not
only are we reviewing almost all of Chapter 4, we are reviewing the
"nuts and bolts" of working with fractions.
5.1 Simplifying Rational Expressions
Rational Expression: An expression that can
be written as a quotient of polynomials. If we use capital letters
to represent polynomials (P & Q), then a rational expression can be
written in the form P/Q where Q does not equal "0."
Rational expressions are fractions that have numerators
and denominators that are polynomials. Everything that is true for
a fraction is true for a rational expression. Recall that we never
have a denominator of "0" -- a quotient with a denominator of
zero is undefined; it is not a real number.
Values Where a Rational Expression is Undefined. Because
division by zero is undefined, any value that causes the polynomial in
the denominator of a rational expression to be "0" results in
a rational expression that is undefined.
Fundamental Principle of Rational Expressions. Just
like a fraction, we can always multiply the numerator and denominator of
a rational expression by any polynomial that does not equal zero and the
value of the product does not change. If P, Q, & R are
polynomials and Q & R do not equal zero; then (PR)/(QR) = P/Q.
Simplifying Rational Expressions:
Factor numerators and denominators.
Apply the fundamental principal to divide out common
factors -- cancel out factors that appear in BOTH the numerator and
Please refer to Examples 3 - 7 (pages 351-352 in text)
to see how this procedures are applied.
5.2 Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions:
Factor numerators and denominators.
Multiply numerators and multiply denominators.
Write product in lowest terms.
While these steps, as outlined in our text, are correct,
if you look at the examples in the book, you will see that factoring
numerators and denominators will allow us to look for common factors.
Step 2 actually just involves writing our factors from step one as a
single denominator!!! Common factors in our numerator AND denominator
can be eliminated, which will greatly simplify our work.
Please be sure to carefully look at Examples 1-3 (pages
357-358 in text). You will see that our objective is to factor the
numerators and denominators completely and cancel out any common
factors. IN MANY WAYS, THIS IS THE "HEART OF ALGEBRA!"
Recall that our definition of algebra stated that
letters represent numbers, that we take "expressions" apart
and then put them back together. DO YOU SEE THAT IN CHAPTER 5 WE
ARE DOING ALL OF THESE THINGS?
After completely factoring numerators and denominators,
canceling out common factors, we simplify what is left to be sure that
the product is written in lowest terms. Canceling out common
factors is really just applying the Fundamental Principal of
Please recall from our review of fractions that WE NEVER
ACTUALLY DIVIDE FRACTIONS.
5.3 Adding and Subtracting Rational Expressions
with the Same Denominator and Least Common Multiple
We can think of rational expressions as being fractions
that are expressed as polynomials in the numerator and denominator.
EVERYTHING WE KNOW ABOUT FRACTIONS APPLIES TO RATIONAL EXPRESSIONS.
This is where I most appreciate E. Elayn Martin Gay's presentation.
If we understood fractions and the foundation that
Chapter 4 was build in (factoring polynomials), we actually have all of
the "tools" we need to successfully work with rational
expressions. Recall that we add fractions with the same
denominator by writing our numerators as addition on the common
denominator. Then, we simply combine the terms.
Remember, we are using capital letters to represent
polynomials. Algebraically (P/R)+(Q/R)=(P+Q)/R (as long as R does
not equal zero). Likewise, (P/R)-(Q/R)=(P-Q)/R.
Recall that we add polynomials by combining like terms.
REMEMBER, THAT WHEN WE SUBTRACT POLYNOMIALS, WE NEED TO CHANGE ALL SIGNS
IN THE SECOND POLYNOMIAL (the one we are subtracting) and then we simply
combine like terms.
We need to always simply our answers by being sure that
all common factors are eliminated -- common factors in the numerator and
denominator "cancel" each other. Be sure to look at
Examples 1-3 (pages 367-368 in text).
If we make the "connections" between our
understanding of fractions and rational expressions, adding/subtracting
polynomials with common denominators is nothing new -- BE SURE TO
SIMPLIFY YOUR ANSWER (see section 5.1).
Our author wants to take advantage of this -- she will
introduce the next unit here so that we have practiced some useful
skills BEFORE we actually need them.
Least Common Multiple
Recall that in Chapter R we discussed how it is not
possible to combine fractions without common denominators. We
needed to find a least common denominator (LCD) -- perhaps you also
remember that LCD=LCM.
On page 368 is a review where we add 8/3 and 2/5.
The common denominator would be 15 -- because creating fractions with
HTML on this Website is time consuming, I will ask you to look at how
the book reviews that example.
We can also find the least common denominator (LCD) for
rational expressions. The least common denominator (*LCD) of a
list of rational exrp3esions is a polynomial of least degree whose
factors include all the factors of the denominator on the list.
LCD of a list of rational expressions:
- Factor each denominator completely.
- The least common denominator (LCD) is the product off all unique
factors found in step 1, each raised to a power equal to the
greatest number of times that the factor appears in any one factored
Please be sure to look at Examples 4-10 (pages 368-370 in text).
5.4 Adding and Subtracting Rational Expressions with Different
We have already reviewed finding the LCD, recall from fractions, that
is the key to adding fractions with different denominators. If we
rewrite fractions as equivalent fractions using the lowest common
denominator (LCD), then we can rewrite our fractions on that common
denominator and combine the numerators.
Add or Subtract Rational Expressions with Different
Find the LCD of the rational expressions (end of
Rewrite each rational expression as an equivalent
expression whose denominator is the LCD found in Step 1.
Add or subtract numerators and write the sum or
difference over the common denominator (at this point, we are really
just combining terms on the common denominator).
Simplify or write in lowest terms (if there are any
common factors in the numerator and denominator, be sure to remove
eliminate them, they "cancel out."
Please review examples 1-7 (pages 375-376).
5.5 Solving Equations Containing Rational Expressions
Before we start, lets think about this -- rational
expression will be more complex to work with than the linear equations
we solved in Chapter 3. LET'S GIVE OURSELVES A BREAK AND USE
ALGEBRA TO ELIMINATE THE ALGEBRA. This will allow us to solve
equations containing rational expressions like we solve any other
Recall that we can always multiply the left and the
right of an equation by the same term and we will not change that
equation's identity. If we apply that principle here, then we will
be ready to solve the equation that remains.
Solve equation Containing Rational Expressions
Multiply both sides of the equation by the LCD of
ALL rational expressions in the equation. THIS WILL ALLOW US
TO "CANCEL OUT" OR ELIMINATE THE DENOMINATOR.
Remove any grouping symbols and solve the resulting
Check the solution IN THE ORIGINAL EQUATION.
Please be sure to look at Examples 1-7 (pages 383-386 in
5.6 Rational Equations and Problem Solving
Recall that all problem solving is presented in our text
as a 4 part process:
Understand. Read and reread the
Translate. Write an equation based on
the words in the problem.
Solve. Apply the steps to solving an
equation to solve for your variable(s).
Interpret. State your answer in a
manner that fully answers the original question.
A some important examples to look at include: