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:

  1. Factor numerators and denominators.

  2. Apply the fundamental principal to divide out common factors -- cancel out factors that appear in BOTH the numerator and denominator.

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:

  1. Factor numerators and denominators.

  2. Multiply numerators and multiply denominators.

  3. 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 Rational Expressions.

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:

  1. Factor each denominator completely.
  2. 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 denominator.

Please be sure to look at Examples 4-10 (pages 368-370 in text).

5.4 Adding and Subtracting Rational Expressions with Different Denominators

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 Denominators

  1. Find the LCD of the rational expressions (end of unit 5.3).

  2. Rewrite each rational expression as an equivalent expression whose denominator is the LCD found in Step 1.

  3. 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).

  4. 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 equation.

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

  1. 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.

  2. Remove any grouping symbols and solve the resulting equation.

  3. Check the solution IN THE ORIGINAL EQUATION.

Please be sure to look at Examples 1-7 (pages 383-386 in text). 

5.6  Rational Equations and Problem Solving

Recall that all problem solving is presented in our text as a 4 part process:

  1. Understand.  Read and reread the problem.

  2. Translate.  Write an equation based on the words in the problem.

  3. Solve.  Apply the steps to solving an equation to solve for your variable(s).

  4. Interpret.  State your answer in a manner that fully answers the original question.

A some important examples to look at include:

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Finding work rates (example 2, page 395-396)

bullet

Solving problems about distance, rate and time ((example 3, pages 295-297)

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Solving problems about similar triangles (example 4, pages 397-398)

5.7 Simplifying Complex Fractions

A rational expression whose numerator or denominator of both contain fractions is called a complex rational expression or complex fraction.  Like ALL fractions, we need to write complex fractions in simplest form.  If we use P and Q to represent polynomials, we have reduced complex fractions to simplest form when P/Q share no common factor.

Our text presents two ways to solve complex fractions -- PLEASE BE SURE TO STUDY AND PRACTICE BOTH.  Like most "choices" we have looked at, there is no "best" way.  Each of us needs to determine what we are most comfortable and accurate -- often, the best method depends on the nature of a problem that is to be solved.

The author simply calls these 2 methods, Method 1 & Method 2.

Method 1:  To simplify a complex fraction:

  1. Add or subtract the fraction in the numerator or denominator so that the numerator is a simple fraction and the denominator is a single fraction.

  2. Perform the indicated division by multiplying the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction.

  3. Write the rational expression in lowest terms.

Please be sure to look at examples 1-3, pages 405-406.

Method 2:  To simplify a complex fraction:

  1. Find the LCD of all the fractions in the complex fraction.

  2. Multiply both the numerator and the denominator of the complex fraction by the LCD from Step 1.

  3. Perform the indicated operations and write the results in lowest terms.

Please be sure to look at examples 4-6, pages 407-408.

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Class Notes

Review Notes

Chapter 1 Notes

Chapter 2 Notes

Chapter 3 Notes

Chapter 4 Notes

Chapter 5 Notes

Chapter 6 Notes

Chapter 7 Notes

Chapter 8 Notes

Chapter 9 Notes

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