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