From 6bf4b90c90f15f4ab60833bddf5b5756d1a6b1f6 Mon Sep 17 00:00:00 2001
From: Elizabeth Alexander Hunt <me@liz.coffee>
Date: Thu, 2 Jul 2026 11:55:17 -0700
Subject: Init

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+% Created 2023-03-01 Wed 08:52
+% Intended LaTeX compiler: pdflatex
+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{graphicx}
+\usepackage{longtable}
+\usepackage{wrapfig}
+\usepackage{rotating}
+\usepackage[normalem]{ulem}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{capt-of}
+\usepackage{hyperref}
+\notindent \notag \usepackage{ dsfont } \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
+\author{Lizzy Hunt}
+\date{\today}
+\title{Assignment Five}
+\hypersetup{
+ pdfauthor={Lizzy Hunt},
+ pdftitle={Assignment Five},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 28.2 (Org mode 9.6.1)}, 
+ pdflang={English}}
+\begin{document}
+
+\maketitle
+\setlength\parindent{0pt}
+
+\section{Section 4.1}
+\label{sec:orge1c762a}
+\subsection{Question One}
+\label{sec:org6594d20}
+\subsubsection{d}
+\label{sec:orgf9d77b6}
+\(2x^5 + x^4 + 6x^2 + 3x + 2\)
+
+\subsection{Question Three}
+\label{sec:org8e6afc5}
+\subsubsection{b}
+\label{sec:org7f9f78d}
+\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\textsuperscript{2} + 1x + 0',
+ '0x\textsuperscript{2} + 1x + 1',
+ '0x\textsuperscript{2} + 1x + 2',
+ '0x\textsuperscript{2} + 2x + 0',
+ '0x\textsuperscript{2} + 2x + 1',
+ '0x\textsuperscript{2} + 2x + 2',
+ '1x\textsuperscript{2} + 0x + 0',
+ '1x\textsuperscript{2} + 0x + 1',
+ '1x\textsuperscript{2} + 0x + 2',
+ '1x\textsuperscript{2} + 1x + 0',
+ '1x\textsuperscript{2} + 1x + 1',
+ '1x\textsuperscript{2} + 1x + 2',
+ '1x\textsuperscript{2} + 2x + 0',
+ '1x\textsuperscript{2} + 2x + 1',
+ '1x\textsuperscript{2} + 2x + 2',
+ '2x\textsuperscript{2} + 0x + 0',
+ '2x\textsuperscript{2} + 0x + 1',
+ '2x\textsuperscript{2} + 0x + 2',
+ '2x\textsuperscript{2} + 1x + 0',
+ '2x\textsuperscript{2} + 1x + 1',
+ '2x\textsuperscript{2} + 1x + 2',
+ '2x\textsuperscript{2} + 2x + 0',
+ '2x\textsuperscript{2} + 2x + 1',
+ '2x\textsuperscript{2} + 2x + 2'\}
+
+\subsection{Question Five}
+\label{sec:orgf501bb3}
+Unfortunately, latex'ing long division is not trivial. Sorry in advance.
+
+\begin{center}
+\includegraphics[width=400px]{./q5.jpeg}
+\end{center}
+
+\subsection{Question Six}
+\label{sec:orgf9d6188}
+\subsubsection{c}
+\label{sec:org0feefbc}
+No, since two polynomials of degree \(\leq k\) with \(k=2\), under multiplication, could produce
+a polynomial of degree 4.
+\subsubsection{d}
+\label{sec:org18fd125}
+\begin{itemize}
+\item Closed under addition (two polynomials with even degrees can only add to polynomials with even degrees)
+\item Closed under multiplication (two polynomials with even degrees can only multiply to polynomials with even degrees)
+\item 0\textsubscript{R} exists in the set
+\item For a polynomial \(a\) with only even powers, then \(a + x = 0_R\) implies that \(a = -x\) which will only change coefficients
+\end{itemize}
+
+Seems like a subring (although not rigorously shown) to me!
+
+\subsubsection{e}
+\label{sec:org6de4719}
+Not a subring, since \(x^3 \cdot x = x^4\), with \(x^3\) and \(x\) both elements in the described set.
+\subsection{Question Eleven}
+\label{sec:orgb3179d8}
+
+\begin{verbatim}
+>>> list(filter(lambda x: ((3 + x) % 9 == 0) and ((3 * x) % 9 == 0), range(9)))
+[6]
+\end{verbatim}
+
+\((1 + 3x)(1 + 6x) = 1 + 6x + 3x + 18x^2 = 1 + 9x + 18x^2 = 1\)
+\end{document}
\ No newline at end of file
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