#+TITLE: Assignment Five
#+AUTHOR: Lizzy Hunt
#+STARTUP: entitiespretty fold inlineimages
#+LATEX_HEADER: \notindent \notag \usepackage{ dsfont } \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+OPTIONS: toc:nil

* Section 4.1
** Question One
*** d
$2x^5 + x^4 + 6x^2 + 3x + 2$

** Question Three
*** b
\begin{verbatim}
>>> list(filter(lambda x: x, \
    [f"{a}x^2 + {b}x + {c}" if not a == 0 or not b == 0 else "" \
    for a in range(3) for b in range(3) for c in range(3)]))
\end{verbatim}

{'0x^2 + 1x + 0',
 '0x^2 + 1x + 1',
 '0x^2 + 1x + 2',
 '0x^2 + 2x + 0',
 '0x^2 + 2x + 1',
 '0x^2 + 2x + 2',
 '1x^2 + 0x + 0',
 '1x^2 + 0x + 1',
 '1x^2 + 0x + 2',
 '1x^2 + 1x + 0',
 '1x^2 + 1x + 1',
 '1x^2 + 1x + 2',
 '1x^2 + 2x + 0',
 '1x^2 + 2x + 1',
 '1x^2 + 2x + 2',
 '2x^2 + 0x + 0',
 '2x^2 + 0x + 1',
 '2x^2 + 0x + 2',
 '2x^2 + 1x + 0',
 '2x^2 + 1x + 1',
 '2x^2 + 1x + 2',
 '2x^2 + 2x + 0',
 '2x^2 + 2x + 1',
 '2x^2 + 2x + 2'}

** Question Five
Unfortunately, latex'ing long division is not trivial. Sorry in advance.

#+attr_latex: :width 400px
[[./q5.jpeg]]

** Question Six
*** c
No, since two polynomials of degree $\leq k$ with $k=2$, under multiplication, could produce
a polynomial of degree 4.
*** d
+ Closed under addition (two polynomials with even degrees can only add to polynomials with even degrees)
+ Closed under multiplication (two polynomials with even degrees can only multiply to polynomials with even degrees)
+ 0_R exists in the set
+ For a polynomial $a$ with only even powers, then $a + x = 0_R$ implies that $a = -x$ which will only change coefficients

Seems like a subring (although not rigorously shown) to me!

*** e
Not a subring, since $x^3 \cdot x = x^4$, with $x^3$ and $x$ both elements in the described set.
** Question Eleven

\begin{verbatim}
>>> list(filter(lambda x: ((3 + x) % 9 == 0) and ((3 * x) % 9 == 0), range(9)))
[6]
\end{verbatim}

$(1 + 3x)(1 + 6x) = 1 + 6x + 3x + 18x^2 = 1 + 9x + 18x^2 = 1$