| Version 2 |
Version 1 |
| In this entry we treat two cases of long division. |
In this entry we treat two cases of long division. |
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| \section{Integers} |
\section{Integers} |
| \begin{thm}[Integer Long Division] |
\begin{thm}[Integer Long Division] |
| For every pair of integers $a, b\neq 0$ there exist a unique integers $q$ and $r$ such that: |
For every pair of integers $a, b\neq 0$ there exist a unique integers $q$ and $r$ such that: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $a=b\cdot q + r,$ |
\item $a=b\cdot q + r,$ |
| \item $0\leq r < |b|$. |
\item $0\leq r < |b|$. |
| \end{enumerate} |
\end{enumerate} |
| \end{thm} |
\end{thm} |
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| \begin{exa} |
\begin{exa} |
| Let $a=10$ and $b=-3$. Then $q=-3$ and $r=1$ correspond to the long division: |
Let $a=10$ and $b=-3$. Then $q=-3$ and $r=1$ correspond to the long division: |
| $$10=(-3)\cdot(-3)+1.$$ |
$$10=(-3)\cdot(-3)+1.$$ |
| \end{exa} |
\end{exa} |
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| \begin{defn} |
\begin{defn} |
| The number $r$ as in the theorem is called the remainder of the division of $a$ by $b$. The numbers $a,\ b$ and $q$ are called the dividend, divisor and quotient respectively. |
The number $r$ as in the theorem is called the remainder of the division of $a$ by $b$. The numbers $a,\ b$ and $q$ are called the dividend, divisor and quotient respectively. |
| \end{defn} |
\end{defn} |
|
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| \section{Polynomials} |
\section{Polynomials} |
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| \begin{thm}[Polynomial Long Division] |
\begin{thm}[Polynomial Long Division] |
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Let $R$ be a commutative ring with non-zero unity and let $a(x)$ and $b(x)$ be two polynomials in $R[x]$, where the leading coefficient of $b(x)$ is a unit of $R$. Then there exist unique polynomials $q(x)$ and $r(x)$ in $R[x]$ such that:
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Let $R$ be a ring and let $a(x)$ and $b(x)$ be two polynomials in $R[x]$, where the leading coefficient of $b(x)$ is a unit of $R$. Then there exist unique polynomials $q(x)$ and $r(x)$ in $R[x]$ such that:
|
| \begin{enumerate} |
\begin{enumerate} |
| \item $a(x)=b(x)\cdot q(x) + r(x),$ |
\item $a(x)=b(x)\cdot q(x) + r(x),$ |
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\item $0\leq \deg(r(x)) < \deg b(x)$ or $r(x)=0$.
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\item $0\leq \deg(r(x)) < \deg b(x)$$.
|
| \end{enumerate} |
\end{enumerate} |
| \end{thm} |
\end{thm} |
|
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| \begin{exa} |
\begin{exa} |
| Let $R=\Ints$ and let $a(x)=x^3+3$, $b(x)=x^2+1$. Then $q(x)=x$ and $r(x)=-x+3$, so that: |
Let $R=\Ints$ and let $a(x)=x^3+3$, $b(x)=x^2+1$. Then $q(x)=x$ and $r(x)=-x+3$, so that: |
| $$x^3+3=x(x^2+1)-x+3.$$ |
$$x^3+3=x(x^2+1)-x+3.$$ |
| \end{exa} |
\end{exa} |
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| \begin{exa} |
\begin{exa} |
| The theorem is not true in general if the leading coefficient of $b(x)$ is not a unit. For example, if $a(x)=x^3+3$ and $b(x)=3x^2+1$ then there are no $q(x)$ and $r(x)$ {\bf with coefficients in} $\Ints$ with the required properties. |
The theorem is not true in general if the leading coefficient of $b(x)$ is not a unit. For example, if $a(x)=x^3+3$ and $b(x)=3x^2+1$ then there are no $q(x)$ and $r(x)$ {\bf with coefficients in} $\Ints$ with the required properties. |
| \end{exa} |
\end{exa} |
