1. C ollege A lgebra Basic Algebraic Operations (Appendix A) L:4 Instructor: Eng. Ahmed Abo absa University of Palestine IT-College

2. 2 Objectives Appendix A Cover the topics in Appendix A (A 4): A-4 : Rational Expressions : basic operations. Multiplying and Dividing Rational Expressions Complex Rational Expressions

3. 3 A-4 A-4 : Learning Objectives After completing this Section, you should be able to: 1.Multiply rational expressions. 2.Divide rational expressions. 3.Simplify complex fractions. A-4 : Rational Expressions: basic operations

4. 4 A-4 A-4 : Multiplying Rational Expressions Multiplying Rational Expressions Q and S do not equal 0.

5. 5 A-4 A-4 : Multiplying Rational Expressions Step 1: Factor both the numerator and the denominator. If you need a review on factoring, feel free to go back to Factoring Polynomials. Step 2: Write as one fraction. Write it as a product of the factors of the numerators over the product of the factors of the denominators. DO NOT multiply anything out at this point. Step 3: Simplify the rational expression. If you need a review on factoring, feel free to go back to Simplifying Rational Expressions. Step 4: Multiply any remaining factors in the numerator and/or denominator. If you need a review on multiplying polynomials, feel free to go back to Polynomials.

6. 6 A-4 A-4 : Multiplying Rational Expressions Step 1: Factor both the numerator and the denominator AND Step 2: Write as one fraction. Example 1: Multiply *Factor the num. and den. In the numerator we factored a difference of squares. In the denominator we factored a GCF and a trinomial.

7. 7 A-4 A-4 : Multiplying Rational Expressions Step 3: Simplify the rational expression. AND Step 4: Multiply any remaining factors in the numerator and/or denominator. *Simplify by div. out the common factors of (y + 3), (y - 3) and y *Excluded values of the original den.

8. 8 A-4 A-4 : Multiplying Rational Expressions Step 1: Factor both the numerator and the denominator AND Step 2: Write as one fraction. Example 2: Multiply *Factor the num. and den. In the numerator we factored a difference of squares. In the denominator we factored a GCF and a trinomial.

9. 9 A-4 A-4 : Multiplying Rational Expressions Step 3: Simplify the rational expression. AND Step 4: Multiply any remaining factors in the numerator and/or denominator. *Simplify by div. out the common factors of (x - 3), 2, and (x + 2) *Excluded values of the original den.

10. 10 A-4 A-4 : Dividing Rational Expressions Dividing Rational Expressions where Q, S, and R do not equal 0. Step 1: Write as multiplication of the reciprocal. Step 2: Multiply the rational expressions as shown above.

11. 11 A-4 A-4 : Dividing Rational Expressions Example 3: Divide *Rewrite as mult. of reciprocal *Factor the num. and den. *Simplifyby div. out the common factors of 3x and (x + 6) *Multiply the den. out *Excluded values of the original den. of product

12. 12 A-4 A-4 : Dividing Rational Expressions Example 4: Divide *Rewrite as mult. of reciprocal *Factor the num. and den. *Simplifyby div. out the common factors of y, (y + 4), and (y - 4) *Multiply the num. and den. out *Excluded values of the original den. of quotient & product

13. 13 A-4 A-4 : Complex Fraction A complex fraction is a rational expression that has a fraction in its numerator, denominator or both. In other words, there is at least one small fraction within the overall fraction. Some examples of complex fractions are: There are two ways that you can simplify complex fractions. We will call them method I and method II.

14. 14 A-4 A-4 : Complex Fraction Method I Simplifying a Complex Fraction Step 1: If needed, rewrite the numerator and denominator so that they are each a single fraction. In other words, you will be combining all the parts of the numerator to form one fraction and all of the parts of the denominator to form another fraction. If you need a review on adding and subtracting rational expressions, go to Adding and Subtracting Rational Expressions.rewrite the numerator and denominator so that they are each a single fraction.Adding and Subtracting Rational Expressions. Step 2: Divide the numerator by the denominator. A fraction bar is the same as a division bar where you are taking the numerator divided by the denominator. Recall that when you divide by a fraction it is the same as multiplying by its reciprocal. So in this step you will be multiplying the numerator by the reciprocal of the denominator. If you need a review on dividing rational expressions, go to Multiplying and Dividing Rational Expressions.Divide the numerator by the denominator.Multiplying and Dividing Rational Expressions.

15. 15 A-4 A-4 : Complex Fraction Method I: Simplifying a Complex Fraction Step 1: If needed, rewrite the numerator and denominator so that they are each a single fraction. In other words, you will be combining all the parts of the numerator to form one fraction and all of the parts of the denominator to form another fraction. If you need a review on adding and subtracting rational expressions, go to Adding and Subtracting Rational Expressions.rewrite the numerator and denominator so that they are each a single fraction.Adding and Subtracting Rational Expressions. Step 2: Divide the numerator by the denominator. A fraction bar is the same as a division bar where you are taking the numerator divided by the denominator. Recall that when you divide by a fraction it is the same as multiplying by its reciprocal. So in this step you will be multiplying the numerator by the reciprocal of the denominator. If you need a review on dividing rational expressions, go to Multiplying and Dividing Rational Expressions.Divide the numerator by the denominator.Multiplying and Dividing Rational Expressions. Step 3: If needed, simplify the rational expression. If you need a review on simplifying rational expressions, go to Simplifying Rational Expressions.simplify the rational expression.Simplifying Rational Expressions.

16. 16 A-4 A-4 : Complex Fraction Example 1: Simplify *Rewrite fractions with LCD of ab Step 1: If needed, rewrite the numerator and denominator so that they are each a single fraction. Combining only the numerator we get:

17. 17 A-4 A-4 : Complex Fraction *Rewrite fractions with LCD of a^2 b Combining only the denominator we get: *Rewrite fractions with LCD of a^2 b Combining only the denominator we get:

18. 18 A-4 A-4 : Complex Fraction Putting these back into the complex fraction we get: Step 2: Divide the numerator by the denominator AND Step 3: If needed, simplify the rational expression. *Write numerator over denominator Putting these back into the complex fraction we get:

19. 19 A-4 A-4 : Complex Fraction *Rewrite div. as mult. of reciprocal *Divide out a common factor of ab *Excluded values of the original den. Putting these back into the complex fraction we get: Note that the value that would be excluded from the domain is 0. This is the value that makes the original denominator equal to 0. Putting these back into the complex fraction we get:

20. 20 A-4 A-4 : Complex Fraction Example 2: Simplify *Rewrite fractions with LCD of (x - 4) Step 1: If needed, rewrite the numerator and denominator so that they are each a single fraction. Combining only the numerator we get:

21. 21 A-4 A-4 : Complex Fraction *Rewrite fractions with LCD of (x - 4) Combining only the denominator we get:

22. 22 A-4 A-4 : Complex Fraction *Write numerator over denominator Putting these back into the complex fraction we get: Step 2: Divide the numerator by the denominator AND Step 3: If needed, simplify the rational expression.

23. 23 A-4 A-4 : Complex Fraction *Rewrite div. as mult. of reciprocal *Divide out a common factor of (x - 4) *Excluded values of the original den. Putting these back into the complex fraction we get: Note that the values that would be excluded from the domain are 4 and 16/5. These are the values that make the original denominators equal to 0.

24. 24 A-4 A-4 : Complex Fraction Method II Simplifying a Complex Fraction Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions. Recall that when you multiply the exact same thing to the numerator and the denominator, the result is an equivalent fraction. If you need a review on finding the LCD, go back to Adding and Subtracting Rational Expressions.the LCD of the smaller fractions.Adding and Subtracting Rational Expressions. Step 2: If needed, simplify the rational expression.simplify the rational expression.

25. 25 A-4 A-4 : Complex Fraction Example 3: Simplify Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions. The denominator of the numerator’s fraction has the following factor: The denominator of the denominator’s fraction has the following factor: Putting all the different factors together and using the highest exponent, we get the following LCD for all the small fractions:

26. 26 A-4 A-4 : Complex Fraction Multiplying numerator and denominator by the LCD we get: *Mult. num. and den. by (x + 1)(x – 1)

27. 27 A-4 A-4 : Complex Fraction Step 2: If needed, simplify the rational expression. *Divide out the common factor of x *Excluded values of the original den. Note that the values that would be excluded from the domain are -1, 1, and 0. These are the values that make the original denominator equal to 0.

28. 28 A-4 A-4 : Complex Fraction Example 4: Simplify Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions. The denominator of the numerator’s fraction has the following factor: The denominator of the denominator’s fraction has the following factor: Putting all the different factors together and using the highest exponent, we get the following LCD for all the small fractions:

29. 29 A-4 A-4 : Complex Fraction Multiplying numerator and denominator by the LCD we get: *Mult. num. and den. by (x + 5)(x - 5) *Excluded values of the original den.

30. 30 A-4 A-4 : Complex Fraction Note that the values that would be excluded from the domain are -5, 5, and - 13/4. These are the values that make the original denominators equal to 0. Step 2: If needed, simplify the rational expression. This rational expression cannot be simplified down any farther