1. Mathematics Relations and Functions: Factoring Polynomials Science and Mathematics Education Research Group Supported by UBC Teaching and Learning Enhancement Fund 2012-2015 Department of Curriculum and Pedagogy FACULTY OF EDUCATION a place of mind

2. Factoring Polynomials I Retrieved from http://catalog.flatworldknowledge.com/bookhub/reader/6329?e=fwk-redden-ch06_s01

3. Multiplying Polynomials I

4. Solution

5. Solution Cont’d

6. Factoring Polynomials II

7. Solution

8. Factoring Polynomials III

9. Solution

10. Solution Cont’d Retrieved from http://www.math10.ca/lessons/polynomials/greatestCommonFactor/greatestCommonFactor.php

11. Factoring Polynomials IV

12. Solution (x + ) ____ + ____ = b ____ × ____ = c

13. Solution Cont’d ____ + ____ = 5 ____ × ____ = -6 6 6

14. Factoring Polynomials V

15. Solution

16. Solution Cont’d In order to get two numbers that multiply into a negative number, these two numbers must have different sign (+,- or -, +) ! Then, for our two numbers, we have (+1, -6), (-1, +6), (+2, -3), or (-2, +3), However, when they are added, we get -6, 6, -1, or 1, respectively. Since our b is -7, we do not have any choice!. Notice that option E does add up to -7, but it multiples into +6. Thus, we have no solution, and this example cannot be factored further! ____ + ____ = -7 ____ × ____ = -6 ?? ? ?

17. Factoring Polynomials VI

18. Solution

19. Solution Cont’d In order to get two numbers that multiply into a positive number, these two numbers must have same sign (+,+ or -, -) ! Then, for our two numbers, we can have ± 6 and ±6, which must have identical signs (+6 with +6, -6 with -6). However, when they are added, we get +12 or – 12. Since our b is +12, our two numbers are +6 and +6. Then, our factored form will look like (x + 6) (x + 6). Notice that this answer can be written as a complete square form Our answer is E. ____ + ____ = +12 ____ × ____ = +36 66 6 6

20. Factoring Polynomials VII

21. Solution

22. Solution Cont’d In order to get two numbers that multiply into a positive number, these two numbers must have the same sign (+,+ or -, -) ! So, looking at the factors of 24, we can have ± 4 and ±6, which must have identical signs (+4 with +6, -4 with -6). However, when they are added together, we get +10 or – 10. Since our b is -10, our two numbers are -4 and -6. Then, our factored form will look like (x - 4)(x - 6). Multiplying with the GCF, we get 3(x - 4)(x - 6). Our answer is C. -4-6 -4 -6

23. Factoring Polynomials VIII

24. Solution 6 6 1 1

25. Solution Cont’d

26. Factoring Polynomials IX

27. Solution -30 4 4 Careful! It cannot be -6 and -20 (if we multiply them we get +120)

28. Solution Cont’d

29. Factoring Polynomials X

30. Solution ____ × ____ = 225 15

31. Solution Cont’d “Difference of squares ”

32. Factoring Polynomials XI

33. Solution

34. Factoring Polynomials XII

35. Solution

36. Solution Cont’d Retrieved from http://maths.nayland.school.nz/Year_11/AS1.2_Algebra_Methods/12_More_Factorising.htm