Homework for Math 252

Asgt #duepagesproblems
12Monday Nov 15 Fill in the details on the exercise on page 7 of the handout on "What are the Real Numbers?" -- i.e., show that the ordering on the collection of all cuts has the Least Upper Bound Property.
11Friday Oct 29 Arrange these rational functions in increasing order, using the ordering discussed in class: t, 200, 1/200, t/(t2+3), 1/t, 2/t, 1, 0, -1
10Monday Oct 25 Two proofs from field theory.
9Wednesday Oct 18 Find all four roots of x4+4x3-9x2-14x-4=0. (They're not rational, but they're not terribly messy either.)
8Wednesday Oct 13 Find the four fourth roots of 1 and the three cube roots of -1.
7Friday Oct 8 Solve x3-3x+10=0 and x3-6x2+6x-2=0 using Cardano's formula. You will have to transform these equations to get them into a form suitable for using with the p and q that appear in the formula. Your answer should be exact (i.e., square roots and cube roots, not decimals). No complex numbers are needed or wanted for this assignment; just find one real root for each equation.
6Friday Oct 1Lovitt, section 5.11 Exercises 3 and 8 -- find all rational roots
5Wednesday Sept 22page about fields Exercise 2. Show that the multiplicative inverse of any nonzero element of a field is uniquely determined by that element.
4Friday Sept 17Hadlock page 23 Problem 7 -- the answer is no. Show this by finding two members of the set {a + bÖ2 + cÖ3 | a,b,cÎQ} whose product is not in that set.
3Monday Sept 13  Prove that the cube root of 2 is irrational.
2Wednesday Sept 8distributed in class Prove three theorems: [1] Every integer greater than 2 is a leg of some Pythagorean triple. [2] For any number b, only finitely many Pythagorean triples have b as a leg. [3] In any Pythagorean triple, at least one of the legs is a multiple of 3.
1Wednesday Sept 1excerpt from Ore's book List all Pythagorean triples that have hypotenuse less than or equal to 25.

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