多項式200811071636

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已知：
1. 若 $f\left( x \right)=\left( ax-b \right)\times q_{1}\left( x \right)+r_{1}$ ，求 $q_{1},r_{1}\,$
2. 若 $f\left( \frac{x}{a} \right)=\left( x-b \right)\times q_{2}\left( x \right)+r_{2}$ ，求 $q_{2},r_{2}\,$
3. 若 $x\times f\left( x \right)=\left( ax-b \right)\times q_{3}\left( x \right)+r_{3}$ ，求 $q_{3},r_{3}\,$

$\left\{ \begin{align} & q_{1}=\frac{q\left( x \right)}{a} \\ & r_{1}=r \\ \end{align} \right.$
$\left\{ \begin{align} & q_{2}=\frac{q\left( \frac{x}{a} \right)}{a} \\ & r_{2}=r \\ \end{align} \right.$
$\left\{ \begin{align} & q_{3}=\frac{q\left( x \right)\times x+r}{a} \\ & r_{3}=\frac{b\times r}{a} \\ \end{align} \right.$

$f\left( x \right)=\left( x-\frac{b}{a} \right)\times q\left( x \right)+r\Rightarrow a\times f\left( x \right)=a\times \left[ \left( x-\frac{b}{a} \right)\times q\left( x \right)+r \right]$
$=a\times \left( x-\frac{b}{a} \right)\times q\left( x \right)+a\times r\Rightarrow a\times f\left( x \right)=\left( ax-b \right)\times q\left( x \right)+a\times r$
$\Rightarrow f\left( x \right)=\frac{\left( ax-b \right)\times q\left( x \right)+a\times r}{a}=\left( ax-b \right)\times \frac{q\left( x \right)}{a}+r$
$\left\{ \begin{align} & f\left( x \right)=\left( ax-b \right)\times \frac{q\left( x \right)}{a}+r \\ & f\left( x \right)=\left( ax-b \right)\times q_{1}\left( x \right)+r_{1} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & q_{1}=\frac{q\left( x \right)}{a}\# \\ & r_{1}=r\# \\ \end{align} \right.$

$f\left( x \right)=\left( x-\frac{b}{a} \right)\times q\left( x \right)+r\Rightarrow f\left( \frac{x}{a} \right)=\left( \frac{x}{a}-\frac{b}{a} \right)\times q\left( \frac{x}{a} \right)+r$
$=\left( \frac{x-b}{a} \right)\times q\left( \frac{x}{a} \right)+r=\left( x-b \right)\times \frac{q\left( \frac{x}{a} \right)}{a}+r$
$\left\{ \begin{align} & f\left( \frac{x}{a} \right)=\left( x-b \right)\times \frac{q\left( \frac{x}{a} \right)}{a}+r \\ & f\left( \frac{x}{a} \right)=\left( x-b \right)\times q_{2}\left( x \right)+r_{2} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & q_{2}=\frac{q\left( \frac{x}{a} \right)}{a}\# \\ & r_{2}=r\# \\ \end{align} \right.$

$f\left( x \right)=\left( ax-b \right)\times \frac{q\left( x \right)}{a}+r\Rightarrow x\times f\left( x \right)=\left( ax-b \right)\times \frac{q\left( x \right)\times x}{a}+r\times x$
$=\left( ax-b \right)\times \frac{q\left( x \right)\times x}{a}+\left( ax-b \right)\times \frac{r}{a}+\frac{b\times r}{a}$
$=\left( ax-b \right)\times \frac{q\left( x \right)\times x+r}{a}+\frac{b\times r}{a}$
$\left\{ \begin{align} & x\times f\left( x \right)=\left( ax-b \right)\times \frac{q\left( x \right)\times x+r}{a}+\frac{b\times r}{a} \\ & x\times f\left( x \right)=\left( ax-b \right)\times q_{3}\left( x \right)+r_{3} \\ \end{align} \right.\Rightarrow \left\{ \begin{align} & q_{3}=\frac{q\left( x \right)\times x+r}{a}\# \\ & r_{3}=\frac{b\times r}{a}\# \\ \end{align} \right.$