1. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals B.2 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Rational Expressions KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Algebra of rational expressions Reduction of rational expressions Complex rational expressions

2. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition of Rational Expression KFUPM - Prep Year Math Program (c) 2009 All Right Reserved A rational expression in is a ration Where p( x ) and q ( x ) are polynomials in and.

3. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Note: Since an integer is a polynomial of degree zero, every rational number is a rational expression. Other examples of rational expressions are KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

4. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Definition: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved The domain of the rational function consists of all real numbers except those for which q (x ) = 0 Definition The value of the rational expression when x is equal to a is obtained by replacing x by a in R ( x ).

5. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 1 find Solution

6. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 2 Find the domain of Solution The domain of Q( x ) is all real numbers except x = 1 and -1

7. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Reducing Rational Expressions KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Reducingconsists of two steps: Step 1 Factor both Step 2 Divide both by the greatest common factor.

8. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 3 Reduce to lowest terms

9. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

10. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Multiplication and Division of Rational Expressions KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Multiplication: Division:

11. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 4 Perform the indicated operations and reduce the resulting rational expression.

12. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

13. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Addition and Subtraction of Rational Expressions KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Identical denominators Different denominator Least Common Denominator (L.C.D)

14. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 5 Find the L.C.D. Solution

15. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Example 6 Solution Perform the indicated operations and simplify. The L.C.D.= (x – a) ( x + a)

16. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Solution The L.C.D.= 3 x ( x + 1) Example 6. (cont.)

17. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Complex Rational Expressions: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved An algebraic expression which has rational expressions in its numerator or denominator is called a complex rational expression. For example,

18. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Complex Rational Expressions: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved To simplify such a complex expression we multiply both its numerator and denominator by the L.C.D. of all the algebraic expressions involved in the makeup of the complex expression. Example 7 Simplify the following complex rational expressions:

19. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Solution

20. E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals KFUPM - Prep Year Math Program (c) 2009 All Right Reserved Solution