Basic Problems: Don't hand these problems in. You should do them before the Core Problems in the same section, to help your understanding.
Core Problems: Everyone must turn these problems in. Always justify your answer, even if the question does not explicitly say so.
MATH 6000 Problems: These problems are required for MATH 6000 students, and are extra credit for MATH 4000 students.
Basic Problems: Sec. 1.1 # 2
Core Problems: Sec. 1.1 # 4 bdgh, 6 (use a in showing b), 8a
MATH 6000 Problem: Sec. 1.1 # 7
Basic Problems: Sec. 1.2 # 1, 6
Core Problems: Sec. 1.2 # 2, 7, 9, 12, 13
Math 6000 Problem: Sec. 1.2 # 20 (Generalizing the result of Exercise 2 means finding the correct statement and proving it.)
Basic Problems: Sec. 1.3 # 4, 20bd
Core Problems: Sec. 1.3 # 8, 9, 18, 20c, 21b, 29
Math 6000 Problem: Sec. 1.3 # 34
Basic Problems: 1 (for Z mod 7)
Core Problems: Sec. 1.4 # 6 bcd, 7, 8, 12, 13
Math 6000 Problem: Sec. 1.4 # 16 (read the last paragraph of p. 39).
Basic Problems: Sec. 2.3 # 3, 4, 8
Core Problems: Sec. 2.3 # 6e, 9bcg, 13, 15 (Note: don't do 6e by repeatedly using the addition formulas for sin and cos; instead think of a way to use deMoivre's formula. Also, for # 15, draw a picture of the roots of the left hand side and use # 4.)
Math 6000 Problem: Sec. 2.3 # 21
Basic Problems: Sec. 3.1 # 2 a, 10 abc
Core Problems: Sec. 2.4 # 6b. Do this by using the method of proof of the cubic formula, as we did in an example in class. Do not use the cubic formula directly or by any other method (since the answers are given, you could just check it -- the point is for you to run through the method of proof of the cubic formula).
Sec. 3.1 # 2 cd, 10 def, 13, 15, 18 . (Hint for #10de: There is an easy way to tell if a cubic polynomial has a factor (think Cor. 1.5). For #10f you can use Cor. 1.5 for part of the proof, but you have to consider another possibility as well.) (Hint for #15: You can use results about calculating derivatives you learned in calculus here!)
Math 6000 Problem: Sec. 3.1 # 20
Basic Problems: Sec. 3.2 # 1, 5c
Core Problems: Sec. 3.2 #2b, 3 ace, 6c, 11, 14
Math 6000 Problem: Sec. 3.2 # 18
Basic Problems: Sec. 3.3 # 3
Core Problems: Sec. 3.3 # 2 abcdfi, 4, 5, 6a, 7, 8
Math 6000 Problem: Sec. 3.3 # 10
Basic Problems: Sec. 5.1 # 3
Core Problems: Sec. 5.1 # 5, 9, 10, 11bcdefg, 13, 15, 21. Note: #13 and #15 make use of the material in the section from Prop. 1.5 on, which will be discussed on Tuesday (you can read it before then).
Math 6000 Problem: Sec. 5.1 # 23
Core Problems: Sec. 5.1 # 12, 19. Sec. 5.2 # 4
Math 6000 Problem: Sec. 5.1 # 17 (You may use the fact that there exist transcendental numbers (that is, real numbers which are not the root of any polynomial equation with rational coefficients) -- for example, pi and e. (This is discussed in Section 4.2.)
Core Problems: Section 5.2 # 9, 10, 11, 13
Math 6000 Problems: Section 5.2 # 16 (you may assume the result of Exercise 3.3.7), 15 (you can use # 16 here!).
Core Problems: Section 4.1 # 3b, 4bcdf, 14ab, 16a, 17ab. (For 14ab, you will have to look at the end of the section; we'll discuss this next class.)
Math 6000 Problem: Section 4.1 # 18