1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.1 Multiplication and Division with Rational Expressions

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Determine any restrictions on the variable in a rational expression. o Reduce rational expressions to lowest terms. o Multiply rational expressions. o Divide rational expressions.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Introduction to Rational Expressions Rational Expressions A rational expression is an algebraic expression that can be written in the form where P and Q are polynomials and

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Introduction to Rational Expressions Notes Remember, the denominator of a rational expression can never be 0. Division by 0 is undefined.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Finding Restrictions on the Variable Determine what values of the variable, if any, will make the rational expression undefined. (These values are called restrictions on the variable.) Solution Set the denominator equal to 0. Solve the equation.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Finding Restrictions on the Variable (cont.) Thus the expression is undefined for Any other real number may be substituted for x in the expression. We write to indicate the restriction on the variable.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Finding Restrictions on the Variable (cont.) Solution Set the denominator equal to 0. Solve the equation by factoring. Thus there are two restrictions on the variable: 6 and –1. We write x ≠  1, 6.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Finding Restrictions on the Variable (cont.) Solution However there is no real number whose square is  36. Thus there are no restrictions on the variable. Set the denominator equal to 0. Solve the equation.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Introduction to Rational Expressions Notes Comments about the Numerator Being 0 If the numerator of a rational expression has a value of 0 and the denominator is not 0 for that value of the variable, then the expression is defined and has a value of 0. If both numerator and denominator are 0, then the expression is undefined just as in the case where only the denominator is 0.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Introduction to Rational Expressions Summary of Arithmetic Rules for Rational Numbers (or Fractions) A fraction (or rational number) is a number that can be written in the form where a and b are integers and b  0. (Remember, no denominator can be 0.) The Fundamental Principle: The reciprocal of and

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Introduction to Rational Expressions Summary of Arithmetic Rules for Rational Numbers (or Fractions) (cont.) Multiplication: Division: Addition: Subtraction:

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Introduction to Rational Expressions The Fundamental Principal of Rational Expressions If is a rational expression and P, Q, and K are polynomials where Q, K ≠ 0, then

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Reducing Rational Expressions Use the fundamental principle to reduce each expression to lowest terms. State any restrictions on the variable by using the fact that no denominator can be 0. This restriction applies to denominators before and after a rational expression is reduced. Solution Note that x  5 is a common factor. The key word here is factor. We reduce using factors only.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Reducing Rational Expressions (cont.) Solution Reduce. The common factor is x – 4. Note that is the difference of two cubes. Also, note that is not factorable.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Reducing Rational Expressions (cont.) Solution Note that the expression 10  y is the opposite of y  10. When nonzero opposites are divided, the quotient is always  1.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Reducing (or Simplifying) Rational Expressions Opposites in Rational Expressions For a polynomial P, where In particular,

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Reducing (or Simplifying) Rational Expressions Notes COMMON ERROR “Divide out” only common factors.INCORRECT 8 is not a common factor. 3 and are not common factors.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Reducing (or Simplifying) Rational Expressions Notes (cont.)CORRECT 4 is a common factor x  3 is a common factor

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Multiplication with Rational Expressions To multiply any two (or more) rational expressions, 1.Completely factor each numerator and denominator. 2.Multiply the numerators and multiply the denominators, keeping the expressions in factored form. 3. “Divide out” any common factors from the numerators and denominators. Remember that no denominator can have a value of 0.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Multiplication with Rational Expressions If P, Q, R, and S are polynomials and Q, S ≠ 0, then

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Multiplication with Rational Expressions Multiply and reduce, if possible. Use the rules for exponents when they apply. State any restrictions on the variable(s).

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Multiplication with Rational Expressions (cont.)

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Multiplication with Rational Expressions (cont.)

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Multiplication with Rational Expressions Notes As shown in Examples 3c and 3d there may be more than one correct form for an answer. After a rational expression has been reduced, the numerator and denominator may be multiplied out or left in factored form. Generally, the denominator will be left in factored form and the numerator multiplied out. As we will see in the next section, this form makes the results easier to add and subtract. However, be aware that this form is just an option, and multiplying out the denominator is not an error.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Division with Rational Expressions If P, Q, R, and S are polynomials and Q, R, S ≠ 0, then Note that is the reciprocal of

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Division with Rational Expressions Solution Note that in this example we have used the quotient rule for exponents.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Division with Rational Expressions (cont.) Solution

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Division with Rational Expressions (cont.) Solution Remember that you have the option of leaving the numerator and/or denominator in factored form.

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Reduce to lowest terms. State any restrictions on the variables. 1. 2. 3. Perform the following operations and simplify the results. Assume that no denominator has a value of 0. 4. 5.

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems (cont.) 6. 7.

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.2. 3. 4. 5. 16. 7.