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2. Factor the following:

8. Factor into linear factors:

10. What is the remainder when is divided by

11. By the Remainder Theorem, the remainder equals the polynomial evaluated at x = 1. That is, the remainder is equal to

12. Identify the graph of A B C D

13. D AB C

14. D CA B

15. AC D B

16. Identify the graph of the hyperbolic paraboloid A B C D

17. A B C D hyperbolic paraboloid

18. Find the equation of the following ellipse

19. Since 3 2 + 4 2 = 5 2, it immediately follows that the equation of the ellipse is given by

20. Match the equation with its graph and with its name. AB CD1. limaçon 2. lemniscate 3. rose 4. cardiod

21. AB CD1. limaçon 2. lemniscate 3. rose 4. cardiod A – 1 – III B – 4 – IV C – 3 – I D – 2 – II

22. Simplify and write in standard form (2 + 3i)(7 – i) – (35/2 + 3i/2)(1+i)

23. = 14 + 19i – 3i2 – 35/2 – 38/2i – 3i2/2 =14 + 19i + 3 – 35/2 – 19i 0 – 3/2 = 17 – 38/2 = 17 – 16 = 1

24. Simplify and write in standard form

27. By DeMoivre’s Theorem, Thus, in polar form, The argument of (1 + i) is The modulus of (1 + i) is

28. Simplify and write in standard form

29. The modulus of (i) is 1 The argument of (i), which equals (0 + i) is Thus, in polar form, By DeMoivre’s Theorem,

30. Find all real and complex values of x such that

31. The cube roots of unity are given by They are depicted in the plot of the unit circle in the complex plane below. Thus, if the three roots are x 0, x 1 and x 2, then

32. Express (2,1) as a linear combination of u and v when u = (3,2) and v = (–1,1).

33. We must solve the equation (2,1) = a(3,2) + b(–1,1) for a and b. This gives the following system of equations: 3a – b = 2 2a + b = 1 Adding the equations gives 5a = 3 so that a = 3/5 and back substituting gives b = 1 – 2(3/5) = –1/5 Hence, (2,1) = 3/5 u – 1/5 v

34. Express the following base 16 number in base 8 (octal) notation. 29A 16

35. One way to do this is to first convert 29A 16 to base 2: 29A 16 = (0010) 2 (1001) 2 (1010) 2 = 001010011010 2 Since we are converting to base 2 3 = 8, we separate into groups of 3: 001010011010 2 = (001) 2 (010) 2 (011) 2 (010) 2 = 1232 8 Or we could have converted into base 10 then to base 8 as follows: 29A 16 = 2(16 2 ) + 9(16 1 ) + 10(16 0 ) = 666 10 Then 666 / 8 = 83 remainder 2 83 / 8 = 10 remainder 3 10 / 8 = 1 remainder 2 1 / 8 = 0 remainder 1 1232 8

36. Let x 1, x 2 and x 3 be the (real and complex) roots of the cubic polynomial Find the sum x 1 + x 2 + x 3

37. Since x 1, x 2 and x 3 are the (real and complex) roots of the cubic polynomial, then Note that the coefficient of x 2 is the sum of the roots x 1, x 2 and x 3. That is, x 1 + x 2 + x 3 = 0.

38. How many different (sensical or otherwise) words can be formed from the letters in the word: OKLAHOMA

39. There are 8 letters in OKLAHOMA, including 2 O’s, 1 K, 1 L, 2 A’s, 1 H and 1 M. This gives a total number of words equal to

40. Find the determinant of the following matrix:

42. Evaluate:

43. Note that for all n > 2 Therefore, Thus, by the Sandwich Theorem, and, hence,

44. Define If it exists, find

46. If there are n rectangles inscribed between the graph of and the x-axis and the lines x = 1 and x = 4 (as shown below), and for, the width of the k th rectangle equals 3 / n, find where A k is the area of the k th rectangle.

47. This sum is equal to the Riemann Sum of the areas of the rectangles. Thus,

48. Simplify the following infinite continued fraction:

50. Evaluate