餘式的性質

出自高材生

$\begin{align} & \left\{ \begin{align} & R_{a}\left( x \right)=r_{1} \\ & R_{a}\left( y \right)=r_{2} \\ \end{align} \right.\Rightarrow R_{a}\left( r_{1}+r_{2} \right)=R_{a}\left( x+y \right) \\ & \left\{ \begin{align} & R_{a}\left( x \right)=r_{1} \\ & R_{a}\left( y \right)=r_{2} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & x=aq_{1}+r_{1} \\ & y=aq_{2}+r_{2} \\ \end{align} \right. \\ & \Rightarrow x+y=a\left( q_{1}+q_{2} \right)+\left( r_{1}+r_{2} \right) \\ & \Rightarrow \left( r_{1}+r_{2} \right)=x+y-a\left( q_{1}+q_{2} \right) \\ & \Rightarrow R_{a}\left( r_{1}+r_{2} \right)=R_{a}\left( x+y-a\left( q_{1}+q_{2} \right) \right) \\ & \Rightarrow R_{a}\left( r_{1}+r_{2} \right)=R_{a}\left( x+y \right)\# \\ & \left\{ \begin{align} & R_{a}\left( x \right)=r_{1} \\ & R_{a}\left( y \right)=r_{2} \\ \end{align} \right.\Rightarrow R_{a}\left( xy \right)=R_{a}\left( r_{1}r_{2} \right) \\ & \left\{ \begin{align} & R_{a}\left( x \right)=r_{1} \\ & R_{a}\left( y \right)=r_{2} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & x=aq_{1}+r_{1} \\ & y=aq_{2}+r_{2} \\ \end{align} \right. \\ & \Rightarrow xy=\left( aq_{1}+r_{1} \right)\left( aq_{2}+r_{2} \right)=aq_{1}aq_{2}+aq_{1}r_{2}+aq_{2}r_{1}+r_{1}r_{2} \\ & \Rightarrow r_{1}r_{2}=xy-a\left( q_{1}aq_{2}+q_{1}r_{2}+q_{2}r_{1} \right) \\ & \Rightarrow R_{a}\left( r_{1}r_{2} \right)=R_{a}\left( xy-a\left( q_{1}aq_{2}+q_{1}r_{2}+q_{2}r_{1} \right) \right) \\ & \Rightarrow R_{a}\left( r_{1}r_{2} \right)=R_{a}\left( xy \right)\# \\ & R_{a}\left( xy \right)=R_{a}\left( r_{1}r_{2} \right) \\ & R_{a}\left( rr \right)=R_{a}\left( xx \right)\Rightarrow R_{a}\left( r^{2} \right)=R_{a}\left( x^{2} \right) \\ & R_{a}\left( r^{2}r \right)=R_{a}\left( x^{2}x \right)\Rightarrow R_{a}\left( r^{3} \right)=R_{a}\left( x^{3} \right) \\ & \cdots \\ & \Rightarrow R_{a}\left( r^{n} \right)=R_{a}\left( x^{n} \right)\# \\ \end{align}$

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餘式的性質

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