2. Chapter 6 Section 1

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 1 The Fundamental Property of Rational Expressions Find the numerical value of a rational expression. Find the values of the variable for which a rational expression is undefined. Write rational expressions in lowest terms. Recognize equivalent forms of rational expressions. 6.1 2 3 4

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Rational Expressions Examples of rational expressions: Rational expressions cannot have a denominator equal to 0 Slide 6.1-3

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Find the numerical value of a rational expression. Slide 6.1-4

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find the value of the rational expression, when x = 3. Solution: Slide 6.1-5 Evaluating Rational Expressions CLASSROOM EXAMPLE 1

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Find the values of the variable for which a rational expression is undefined. Slide 6.1-6

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The 11 th Commandment Thou shall not… divide by zero The denominator of a rational expression cannot equal 0 because Division by 0 is Undefined For instance, in the rational expression  If x is 2, then the denominator becomes 0, making the expression undefined.  Thus, x cannot equal 2. We indicate this restriction by writing x ≠ 2. Denominator cannot equal 0 Slide 6.1-7

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Finding Restrictions on the Variable Step 1: Set the denominator of the rational expression equal to 0. Step 2: Solve this equation. Step 3: The solutions of the equation are the values that make the rational expression undefined. The variable cannot equal these values. Slide 6.1-8 Determining When a Rational Expression is Undefined

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Find any values of the variable for which each rational expression is undefined. Solution: never undefined Slide 6.1-9 Finding Values That Make Rational Expressions Undefined CLASSROOM EXAMPLE 2

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Write rational expressions in lowest terms. Slide 6.1-10

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write rational expressions in lowest terms. Fundamental Property of Rational Expressions where K ≠ 0 and Q ≠ 0 Lowest Terms If the greatest common factor of its numerator and denominator is 1. This property is based on the identity property of multiplication, since Slide 6.1-11

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Write each rational expression in lowest terms. Slide 6.1-12 Writing in Lowest Terms CLASSROOM EXAMPLE 3

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Only common factors can be divided out, not common addends!!! Step 1: Factor the numerator and denominator completely. Step 2: Use the fundamental property to divide out any common factors. Addends cannot be divided out. Slide 6.1-13 Writing a Rational Expression in Lowest Terms Like This NOT like this!

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Write in lowest terms. Slide 6.1-14 Writing in Lowest Terms CLASSROOM EXAMPLE 4

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Quotient of Opposites If the numerator and the denominator of a rational expression are opposites, as in then the rational expression is equal to −1. Write in lowest terms. Slide 6.1-15 Writing in Lowest Terms (Factors Are Opposites) CLASSROOM EXAMPLE 5

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write each rational expression in lowest terms. or Slide 6.1-16 Writing in Lowest Terms (Factors Are Opposites) Solution: CLASSROOM EXAMPLE 6

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Recognize equivalent forms of rational expressions. Slide 6.1-17

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Recognize equivalent forms of rational expressions Three ways to write the common fraction = = The − sign representing the factor −1 is in front of the expression. = = The factor may instead be placed in the numerator or in the denominator. Slide 6.1-18

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Be careful to apply the distributive property correctly. Choose the equivalent expression. Explain. a. b. Slide 6.1-19 Recognize equivalent forms of rational expressions

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write four equivalent forms of the rational expression. Solution: Slide 6.1-20 Writing Equivalent Forms of a Rational Expression CLASSROOM EXAMPLE 7