HCF & LCM-200811211001

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屬性

$\begin{align} & f\left( x \right)=x^{3}+1 \\ & g\left( x \right)=2x^{3}-3x^{2}+2x+7 \\ \end{align}$

$f\left( x \right)=x^{3}+1^{3}=\left( x+1 \right)\left( x^{2}-x+1 \right)$
$\begin{align} & g\left( x \right)=2x^{3}-3x^{2}+2x+7 \\ & =2x^{3}+2-3x^{2}+2x+5 \\ & =2\left( x^{3}+1 \right)-\left( 3x^{2}-2x-5 \right) \\ & =2\left( x+1 \right)\left( x^{2}-x+1 \right)-\left[ \left( 3x-5 \right)\left( x+1 \right) \right] \\ & =\left( x+1 \right)\left[ 2\left( x^{2}-x+1 \right)-\left( 3x-5 \right) \right] \\ & =\left( x+1 \right)\left( 2x^{2}-2x+2-3x+5 \right) \\ & =\left( x+1 \right)\left( 2x^{2}-5x+7 \right) \\ \end{align}$
$\Rightarrow \left\{ \begin{align} & f\left( x \right)=\left( x+1 \right)\left( x^{2}-x+1 \right) \\ & g\left( x \right)=\left( x+1 \right)\left( 2x^{2}-5x+7 \right) \\ \end{align} \right.$
$H.C.F.\left[ f\left( x \right),g\left( x \right) \right]$ 為 $\left( x+1 \right)\#$
$L.C.M.\left[ f\left( x \right),g\left( x \right) \right]$ 為 $\left( x+1 \right)\left( x^{2}-x+1 \right)\left( 2x^{2}-5x+7 \right)\#$