1. Gerald Chrisman Gateway Technical College 1

2. 2 Algebra II Series of parameter driven, algebra exercises involving factoring polynomial expressions, quadratic equation, and using radical expressions involving square and cube roots.

3. Gerald Chrisman Gateway Technical College 3 Algebra 101 Students more fully understand factoring and the quadratic equation by simplifying equations with radical expressions. The equations contain square roots, cube roots, and squares and cubes of equations are used. The use of a mechanical device, calculator, may be required to check the answers. Students solving these exercises will enhance the understanding of factoring and solutions to equations. In addition the instructor can change the parameters, constants, to make each assignment different. The following three parts characterize three different skill levels.

4. Gerald Chrisman Gateway Technical College 4 PART 0 Factoring

5. Gerald Chrisman Gateway Technical College 5 PART I Instructing students about factoring trinomials as well as squares and cubes of equations is enhanced with exercises that are more challenging. For instance in addition to the usual trinomial factoring problems include expressions such as; which reduces to the quadratic Thus, the student’s algebraic skills are easily evaluated.

6. Gerald Chrisman Gateway Technical College 6 Sample Next introduce equations such as;. Solving for x is a routine matter and the instructor can make many variations for practice. The general solution, using parameters a,b,c follows.

7. Gerald Chrisman Gateway Technical College 7 General solution

8. Gerald Chrisman Gateway Technical College 8 Some examples

9. Gerald Chrisman Gateway Technical College 9 Sample graph

10. Gerald Chrisman Gateway Technical College 10 A

11. Gerald Chrisman Gateway Technical College 11 B

12. Gerald Chrisman Gateway Technical College 12 C

13. Gerald Chrisman Gateway Technical College 13 PART II Next the equation contains square roots that when simplified is a rectangular hyperbola.

14. Gerald Chrisman Gateway Technical College 14 General solution

15. Gerald Chrisman Gateway Technical College 15 Example

16. Gerald Chrisman Gateway Technical College 16 Sample graph

17. Gerald Chrisman Gateway Technical College 17 PART III Next as students become more able to handle square roots introduce the cube root equations such as;

18. Gerald Chrisman Gateway Technical College 18 This equation, when taken to the third power, has an interesting exercise property; it factors like this. Notice the original equation returns in the solution and a quadratic equation results.

19. Gerald Chrisman Gateway Technical College 19

20. Gerald Chrisman Gateway Technical College 20 Sample graph

21. Gerald Chrisman Gateway Technical College 21

22. Gerald Chrisman Gateway Technical College 22 Sample graph

23. Gerald Chrisman Gateway Technical College 23

24. Gerald Chrisman Gateway Technical College 24 Sample graph

25. Gerald Chrisman Gateway Technical College 25 General solution

26. Gerald Chrisman Gateway Technical College 26 Additional examples

27. Gerald Chrisman Gateway Technical College 27

28. Gerald Chrisman Gateway Technical College 28 More Equations with cube roots

29. Gerald Chrisman Gateway Technical College 29 CONCLUSION The above examples provide the instructor with a variety of challenging algebra problems for students with various skill levels in algebra. In addition the parameters provide additional flexibility for instructors to repeat these exercises. These equations and others like them have assisted my classes more fully understand algebra as well as solutions that require mechanical checking, and several steps.

30. Gerald Chrisman Gateway Technical College 30 Thank you Gerald Chrisman Gateway Technical College Racine Campus chrismang@gtc.edu