5Go to the solutions page Consider the polynomial $f(x) = (x-1)^2(x-2)$.
[(1)] Let $g(x)$ be a polynomial with real coefficients, and let $r(x)$ be the remainder when $g(x)$ is divided by $f(x)$. Show that the remainder when $g(x)^7$ is divided by $f(x)$ equals the remainder when $r(x)^7$ is divided by $f(x)$.
[(2)] Let $a$, $b$ be real numbers, and let $h(x) = x^2 + ax + b$. Let $h_1(x)$ be the remainder when $h(x)^7$ is divided by $f(x)$, and let $h_2(x)$ be the remainder when $h_1(x)^7$ is divided by $f(x)$. Find all pairs $a$, $b$ such that $h_2(x)$ equals $h(x)$.
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\noindent\textbf{5} \hfill \textit{Go to the solutions page}
\medskip
\noindent Consider the polynomial $f(x) = (x-1)^2(x-2)$.
\begin{enumerate}
\item[(1)] Let $g(x)$ be a polynomial with real coefficients, and let $r(x)$ be the remainder when $g(x)$ is divided by $f(x)$. Show that the remainder when $g(x)^7$ is divided by $f(x)$ equals the remainder when $r(x)^7$ is divided by $f(x)$.
\item[(2)] Let $a$, $b$ be real numbers, and let $h(x) = x^2 + ax + b$. Let $h_1(x)$ be the remainder when $h(x)^7$ is divided by $f(x)$, and let $h_2(x)$ be the remainder when $h_1(x)^7$ is divided by $f(x)$. Find all pairs $a$, $b$ such that $h_2(x)$ equals $h(x)$.
\end{enumerate}
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