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\begin{center}
{\large{\textsc{Homework 17}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
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Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}

\item Let $a, b, c \in \NN$. Prove that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$, then $\gcd(a,bc) = 1$.

[\textsc{hint:} First check that the statement is true if any of $a$, $b$, or $c$ is equal to $1$. Then, for the case where $a > 1$, $b > 1$, and $c > 1$, consider unique prime factorizations.]


\item Let $a, b \in \NN$ and set $d = \gcd(a,b)$.
\begin{enumerate}
\item Explain why $\frac{a}{d}$ and $\frac{b}{d}$ are integers.
\item Prove that $\gcd\!\big( \frac{a}{d}, \frac{b}{d} \big) = 1$.

[\textsc{hint:} As in the previous problem, there is very little to prove when $a = 1$ or $b = 1$. In the case where $a > 1$ and $b > 1$, look at prime factorizations.]
\end{enumerate}

\end{enumerate}

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\begin{center}
\textbf{Practice Problems}
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It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item \textbf{{[Falkner Section 4 Exercise 20 (The rational roots theorem)]}} Let $r \in \QQ$ be a rational number such that
\[ c_n r^n + c_{n-1} r^{n-1} + \cdots + c_1 r + c_0 = 0, \]
where $n \in \NN$ and $c_0, c_1, \ldots, c_n, c_{n-1} \in \ZZ$, and $c_n \neq 0$. In other words, $r$ is a rational root of the polynomial $f(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0$, and every coefficient of this polynomial is an integer.

Prove that $r$ can be written in the form $r = \frac{a}{b}$. where $a$ is an integer such that $a | c_0$ and $b$ is a nonzero integer such that $b | c_n$.

[\textsc{hint:} Write $r = \frac{a}{b}$ as a fraction in lowest terms. Then $\gcd(a,b) = 1$. (Why?) What does this say about the prime factorizations of $a$ and $b$?]

\item \textbf{{[Falkner Section 4 Exercise 21]}} Let $f(x) = 3x^3 - 40x^2 + 97x + 10$.
\begin{enumerate}
\item Find a rational number $r \in \QQ$ such that $f(r) = 0$. [\textsc{hint:} Use the rational roots theorem to narrow down the possibilities for $r$.]
\item Find two other real numbers $s$ and $t$ such that $f(s) = 0$ and $f(t) = 0$. [\textsc{hint:} Use part (a) and polynomial long division to write $f(x) = (x-r) \, g(x)$, where $g(x)$ is a quadratic polynomial.]
\item Explain why $s$ and $t$ must be irrational. [\textsc{hint:} There are several ways to do this. One elegant way is to notice that $g(x) = 3h(x)$, where $h(x)$ is a quadratic polynomial to which it is particularly easy to apply the rational roots theorem.]
\end{enumerate}
\end{enumerate}




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