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\begin{document}
\title{Math 301 Homework}
\author{Mary Radcliffe}
\date{Due 25 January 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $a, b, c$ be integers. Suppose that $c|a$ and $c|b$. Prove that $c|(a+b)$.

\item Let $a$ and $b$ be irrational numbers, and let $x$ be a rational number. Determine if the following are irrational, rational, or whether it cannot be determined. Prove that your answer is correct.
\begin{enumerate}
\item $a+b$
\item $a+x$
\item $ax$
\item $ab$
\end{enumerate}

\item Let $a, b, r$ be integers, and suppose that $a$ leaves a remainder of $r$ when divided by $b$. Prove that $a$ also leaves a remainder of $r$ when divided by $-b$.

\item Let $a, b \in R$ and let $p(x) = x^2+ax+b$. The value $\Delta = a^2-4b$ is called the discriminant of $p$. Prove that $p$ has no real roots if $\Delta<0$, one real root if $\Delta =0$, and two real roots if $\Delta>0$.

\item Let $p(x)$ be a polynomial with coefficients in $\R$. Suppose that $\alpha$ is a root of $p(x)$. Prove that $\alpha$ is also a root of $(x-a)p(x)$ for all $a\in \R$.

\item Identify which of the following are propositions, and discuss what might be required to prove them if they are.
\begin{enumerate}
\item This is the tallest sheep in the world.
\item My dog Rufus.
\item The square of every even integer is also an even integer.
\item A kitty is orange.
\item This kitty is orange.
\item Go to the store.
\end{enumerate}

\end{enumerate}
\end{document}