Math 3107, Section 2       Monday, April 26

Spring 1999

Miscellaneous exercises - set 4

Part I: Homework due Wednesday, April 28

1.  (This is similar to exercise 10.5.1 in the notes) Determine whether each of the following polynomials is irreducible (i) over the rationals and (ii) over the reals.

  1. x2 - 5x + 4
  2. x2 - x + 2
  3. x4 - 1
  4. x4 - 2

2. (This is similar to exercises 10.5.3 and 10.5.4 in the notes.) Let K = Q(), the set of all numbers of the form a + b , where a and b are rational numbers.

  1. Verify that K is closed under multiplication. Thus, if a + b and c + d are elements of K, calculate their product and verify that it is an element of K.
  2. Calculate the multiplicative inverses of 2 - and 3 + 2 in K.

3. = Exercise 10.5.7 in the notes.
Suggestion: The goal is to find a polynomial with integer coefficients of which the given number is a root. For example, 21/3 = is a root of the polynomial x3 - 2.

4. = Exercise 10.5.8 in the notes.
Suggestion: When calculating ( + )2, be sure to make use of the fact that ()2 = 2 and ()2 = 3
 

Part II: Review questions

Note: This covers some but not all of the topics in chapter 10 of the notes. A couple of other review questions will be presented in class on Monday, if there is time. You should also review past homework problems.

1.

  1. Given that \pi = 3.14159263538979324... , determine the least upper bound of the following set of rational numbers

    2. 3.1, 3.14, 3,141, 3,1415, 3.14159, 3.141592, 3.1415926, 3.14159265, . . .
  2. Does this set have a least upper bound in the set of all rational numbers?

2. Calculate the following complex numbers:

  1. (3 + 5i )·(2 - 3i )
     

  2.  
  3. ,
     
      Here, a, b, c, and d are real numbers
     
  4. The complex conjugate of 
     
3. Let f(x) = 2x2 - 9x + 10.
  1. Given that x = 2 is a solution of the equation f(x) = 0, factor the quadratic polynomial f(x) = 2x2 - 9x + 10 as a product of x - 2 and another first-degree factor.:
     

    2x2 - 9x + 10 = (x - 2)(__x + ___ )
     

  2. Given that x = -2 is a solution of the equation g(x) = 0, factor the third degree polynomial g(x) = x3 + 3x2 - 4 as a product of x + 2 and a quadratic factor:

     

    x3 + 3x2 - 4 = (x + 2) (x2 __x - ___ )
     
     (In addition to finding the coefficient of x, insert the appropriate plus or minus sign.)
     

  3. With g(x) as in part (b), does the equation g(x) = 0 have (i) three real roots, or (ii) one real root and two complex roots?