Math 502
Written Assignment No. 2

due Tuesday, March 4, 2008

Directions: Use Maple for assistance in responding to the following problems. Please typeset your solutions. Explain what you have done. Maple session details are not necessary unless you think it important to include them. Accuracy is important.

Although you may refer to books and notes, you may not seek help from others on this written assignment.

Answer the following questions:

1. For each of the following vector triples $\left[u,v,w\right]$ and integers $b>1$ find the rational number $r$, expressed as the ratio of two integers in base $10$ notation, for which $u$ is the sequence of $b$-adic digits (from left to right) for the integer part of $r$, $v$ is the sequence of $b$-adic digits for the portion of the fractional part of $r$ that precedes the repeating portion, and $w$ is the sequence of $b$-adic digits in the repeating portion of the fractional part of $r$.

   b u v w 10 [5, 1, 8] [8, 6, 7] [5, 3, 0, 9] 16 [B, A, D] [C, 0, D, E] [F, A, D] 9 [3] [0, 4] [2, 0, 0, 8]

2. If $P=a+bx+c{x}^2+d{x}^3+e{x}^4+f{x}^5$ is a polynomial, find polynomials $q_0,q_1,q_2,\dots$ of degree at most $1$ such that $$P=\sum _{j\ge 0}{q}_j{\left(x^2+1\right)}^j\text{.}$$

3. Factor the polynomial $x^{31}-1$ as

   1. a rational polynomial.

   2. a polynomial mod $2$.

   3. a polynomial mod $3$.

   4. a polynomial mod $5$.

4. Find the partial fraction expansion of $$\frac{1}{x^{11}+2x^9+x^7-x^4-2x^2-1}$$ regarded as a “rational function” with:

   1. rational coefficients.

   2. coefficients in the integers mod $2$.

   3. coefficients in the integers mod $7$.

   4. coefficients in the integers mod $11$.

5. The sequence of integers

    [61, 13, 25, 37, 13, 28, 17, 81, 40, 3, 75, 91, 13, 25, 20, 37, 81,

     25, 91, 95, 25, 91, 17, 70, 37, 17, 25, 77, 25, 75, 85, 25, 13, 91,

     3, 25, 19, 75, 20, 37, 19, 81, 25, 3, 91, 25, 23, 37, 25, 73, 37,

     17, 81, 25, 85, 37, 28, 59, 17, 37, 6]

   is the sequence of $17$-th powers mod $97$ of numbers obtained from a sequence of ASCII codes by left-shifting each code 30 units so the sequence occupies the interval $\left[2,96\right]$ instead of the normal printable ASCII interval $\left[32,126\right]$. What text string led to the original sequence?