Lesson 1

Lesson 1: Simplifying Rational Expressions

Focus

Flight simulators are practical training tools for people learning to be pilots. These simulators mimic actual flying conditions. Flight students must control the plane using equipment and instrumentation similar to what is found in real aircraft.

Driving simulators are used in rehabilitation hospitals to help people who have been injured learn how to drive again. Simulators provide a safe environment for learning how to navigate different scenarios that may be encountered in real life. The simulated experience is equivalent to the real one in many ways, but the simulated experience is simplified for training purposes. Can you think of other examples of simulations? What types of situations are being simulated?

In previous math courses you worked with rational numbers. The strategies you used to generate equivalent fractions and to simplify fractions will help you to do the same processes with rational expressions. Your previous experience with rational numbers can be thought of as the training for, or simulation of, the work you will do with rational expressions in this lesson and module.

Outcomes

At the end of this lesson you will be able to

• compare the strategies for writing equivalent forms of rational expressions to the strategies for writing equivalent forms of rational numbers
  
  
• determine the non-permissible values for a rational expression and explain why such values are non-permissible
  
  
• determine equivalent rational expressions to given rational expressions and explain why the non-permissible values of both are the same
  
  
• simplify a rational expression and identify, correct, and explain errors in the simplification of a rational expression

Lesson Questions

You will investigate the following questions:

• How is the process of simplifying rational expressions similar to the process of simplifying rational numbers?
  
  
• Why is simplification desired in mathematics?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 1 Assignment (Download the Lesson 1 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 6 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you are having difficulty with concepts or calculations, contact your teacher.

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

Before beginning this lesson you should be able to

• define rational numbers
  
  
• write equivalent fractions
  
  
• determine the greatest common factor (GCF) of a set of numbers
  
  
• simplify fractions

Are You Ready?

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. What is a rational number? Answer
   
   
2. Complete each statement so that the fractions are equivalent.
   
   
   1. Answer
      
      
   2. Answer
      
      
   3. Answer
      
      
3. Determine the prime factorization of each expression.
   
   
   1. 24 Answer
      
      
   2. 60 Answer
      
      
   3. 30x2y Answer
      
      
4. Simplify the following fractions.
   
   
   1. Answer
      
      
   2. Answer
      
      
   3. Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.

Refresher

Review the definition and work through the demonstration applet in Rational Numbers.

Review examples of writing and simplifying fractions in Equivalent Fractions.

Greatest Common Factor (GCF) provides a definition to review and an applet to explore factors and greatest common factors.

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

© Rob Byron/3683987/Fotolia

In this section you will play a game to discover something about rational expressions.

Try This 1

Follow the steps in Table 1.

1. How does the final result compare to the original number?
   
   

2. Repeat the procedure with a different starting number each time. What do you notice?
   
   
3. Copy Table 1, and add a third column, as shown, called “Expression.” Write an expression for each step in the table beginning with the expression “x”.
   
   
    

   TABLE 1
   Step	Instruction	Expression
   1	Pick any number.	x
   2	Square the number.
   3	Subtract 4 from the result.
   4	Divide the result by two more than the original number.

   
   
   

4. Based on your answers to questions 1 and 2, what is another expression that could be used to represent the final result?
   
   

5. There is one starting number that does not yield a result similar to the other results. Can you determine this number?

Save your responses in your course folder.

You will need to retrieve these results later in the lesson in order to check your work and to share what you have learned with a classmate.

Explore

Do you have a favourite board game? Perhaps you like classic games, such as Monopoly, The Game of Life, or Snakes and Ladders! Two important principles of game design are realism and simplicity. Part of what makes these games enjoyable is the fact that they simulate real-world contexts. In the examples provided here, one is a real-estate trading game, another takes you through the milestones of life, and the third simulates life with ladders rewarding good acts and snakes penalizing bad acts.

What really makes these games fun and engaging is that they portray realistic situations and have rules that young children can understand. For example, gameplay in Monopoly is not bogged down by housing inspections or lawyer fees that would normally come with every real-life property purchase. These aspects, though realistic, would complicate the game and reduce the game’s entertainment value.

In this lesson you will study rational expressions. Just like a game designer creates games based on reality, you will learn how to write algebraic expressions that are equivalent to other expressions. As well, just as games are simplified versions of the real thing, you will learn to simplify rational expressions.

Save Module 6 Glossary Terms in your course folder now.

Rational Expressions

Recall the definition of a rational number: a real number that can be expressed in the form , where a and b are integers and b ≠ 0.

A rational expression has a similar definition. A rational expression can be expressed in the form of , where a and b are polynomials and b ≠ 0.

Retrieve your work from Try This 1. In that activity you picked a number and applied a series of mathematical operations to the number. You should have obtained the following rational expression to describe the result:

You may have noticed that the result is always two less than the original number. For example, if you started with the number 19, your final result was 17. You may have also determined that there was one number that does not follow this pattern.

Share 1

Confirm that x = −2 does not yield the same result as any other value of x. You may want to use a calculator to do this. What does your calculator show when a rational expression is undefined?

With a classmate discuss why x = −2 does not follow the same pattern.

Save your responses in your course folder.

Non-Permissible Values

A rational expression is not defined when the expression’s denominator is zero. Therefore, any value of the variable that would result in a denominator of zero is considered a non-permissible value.

Try This 2

For each of the rational expressions in the table, determine the non-permissible value(s). You can check your answers by using your calculator.

Rational Expression	Non-Permissible Value(s) of x

Save your response in your course folder.

The Property of 1

What happens to a number if you multiply the number by 1? If you said that the number remains unchanged, you are absolutely right. This property of one is critical in determining equivalent expressions in mathematics.

Try This 3

1. Multiply or divide the following numbers.
   
   
   1. 
      
      
   2. 
      
      
   3. 
      
      
2. Verify that the end result for each of the calculations in question 1 is equivalent. You may want to do this by using your calculator to check that the decimal representation of the original number is identical to that of the final number.
   
   
3. To what number are and equivalent?
   
   
4. Compare the following two expressions:
   
   

    

   and
   
   
   1. Identify the non-permissible value(s) for s in each expression.
      
      
   2. Are the expressions equivalent? How do you know?

Save your responses in your course folder.

You will use your responses to question 1 to complete Share 2, Part B.

Generating Equivalent Rational Expressions

Simplifying Rational Expressions

To simplify a rational expression means to write an equivalent expression where the greatest common factor between the numerator and the denominator is 1. Such an expression is described as being in simplified form, or reduced to lowest terms.

The following example shows how you can simplify a rational number by using the method of prime factorization. Pay careful attention to the steps of this example. You will apply these steps to rational expressions in Try This 4.

Example

1. Simplify  by using prime factorization.
   
   
2. Verify that the simplified form is equivalent to the original rational number.

Solution

1. 
   
   By rearranging the order of the factors in the numerator, the expression can be written as the product of three fractions.
   
   
    
   
   
   Since the first two fractions are equal to 1, the expression is simplified to
   
   
    
   
   
   Therefore, according to the property of 1, where any number multiplied by 1 is equal to itself,
   
   
    
   
   
   
2. You can use cross-products to verify that
   
   
    
   
   
   Since the cross-products are equal,  is equivalent to

In Try This 1 you found that by applying a series of mathematical operations to a number, you end up with a number that is two less than the original number. The following statement summarizes this observation.

The expression is equivalent to the expression x − 2  for all values of x, except when x = −2.

In Try This 4 you will use the same principles demonstrated in the previous example to confirm this result.

Try This 4

1. Consider the rational expression Show that this expression simplifies to x − 2.
   
   
   
   
2. Simplify the rational expression State any non-permissible values.

 Save your responses in your course folder.

Share 2

Part A: Discussion

Discuss the following questions with a partner. Record your responses.

1. What are some key differences between simplifying rational numbers and simplifying rational expressions?
   
   
2. Why is it necessary to include the non-permissible value(s) with the simplified form even though this form may not have a denominator?
   
   
3. Based on your answer to question 2, when should you identify non-permissible values?

Part B: Calculator Verification

You can use your graphing calculator to verify that two expressions are equivalent. Follow the steps outlined below for each of three rational expressions and their simplified forms. Use the following rational expressions and the equivalent expressions you obtained in Try This 3 question 1:

Step 1: Input the rational expression into Y1 of the Y= editor.

Step 2: Input the other rational expression into Y2 of the Y= editor. If you have the option to do so, set the graph to a different thickness.

Step 3: Graph the expressions. If the first expression graphs over the second expression, you have confirmed that the expressions are the same.

Discuss the following questions in relation to the calculator verification procedure.

4. The graphs should appear to be identical. In fact, they are not. Can you identify where the two graphs are not identical and give an explanation as to why this is the case?
   
   
5. How can you recognize when two rational expressions that appear to be identical are, in fact, not identical?

Save your responses in your course folder.

Your teacher may wish to check your answers to ensure you have a good grasp of the material learned so far.

Try This 5

Watching for Errors

Thus far in the lesson, you have learned how to write equivalent rational expressions and reduce the expressions to lowest terms. In the remainder of the lesson you will study errors that can occur in the process of simplifying rational expressions. You will do this by analyzing the work of fictional students.

Try This 6

The work of four different students is shown. Each student has made an error in simplifying certain rational expressions.

• Analyze the work of each student.
  
  
• Identify and explain the error.
  
  
• Show the correct solution.

1. Marlee
   
   
    
   
   
   
2. Neetu
   
   
    
   
   
   
3. Aaron
   
   
    
   
   
   
4. Siobhan
   
   
    
   
   

Save your responses in your course folder.

Share 3

With a classmate, compare your answers to Try This 6. Discuss how the errors you identified can be avoided. In other words, are there strategies that can be applied so that the risk of making the mistakes is minimized?

Save your responses in your course folder.

Make any necessary revisions to your work from Try This 6. You will be asked to copy your work from Try This 6 into the Lesson 1 Assignment later in this lesson.

Self-Check 3

Connect

Lesson 1 Assignment

Lesson 1 Summary

You investigated the following questions:

• How is the process of simplifying rational expressions similar to the process of simplifying rational numbers?

• Why is simplification desired in mathematics?

You discovered that simplifying rational expressions is similar to simplifying rational numbers. In both cases, you must find factors common to both the numerator and the denominator. It is appropriate to eliminate these common factors according to the property of 1.

The property of 1 states that the multiplication of any number by 1 is equal to itself. Therefore, eliminating factors common to both the numerator and the denominator results in an equivalent, but simplified, expression.

You discovered a significant difference between simplifying rational numbers and simplifying rational expressions. In the case of rational expressions, you must consider non-permissible values. Any value(s) of the variable that would result in a denominator equal to zero is considered non-permissible.

The process of simplifying is an essential skill that should be part of any math student’s repertoire. Simplifying expressions allows you to simplify calculations, especially those that involve multiple steps. Reducing the size of the numbers and the amount of variables that you must carry from step to step makes subsequent calculations easier and reduces the risk of errors. In the next two lessons you will confirm this claim as you multiply, divide, add, and subtract rational expressions.