增減項因式分解法

佇 Youtube 个 "360 KNOWLEDGE WORLD" 看著即題个因式分解，是用「增減項」个技巧來鬥出會使先佇局部進行因式分解个「場面」, 即種解題法加減著靠靈感佮試誤：

\begin{equation} \begin{aligned} & \kern+0.4em \kern+0.4em \thinspace x^{7} +x^{5} +1\\ & =x^{7} +x^{6} +x^{5} -x^{6} +1\\ & =x^{5}\left( x^{2} +x+1\right) -\left( x^{6} -1\right)\\ & =x^{5}\left( x^{2} +x+1\right) -\left( x^{3} +1\right)\left( x^{3} -1\right)\\ & =x^{5}\left( x^{2} +x+1\right) -\left( x^{3} +1\right)( x-1)\left( x^{2} +x+1\right)\\ & =\left( x^{2} +x+1\right)\left[ x^{5} -\left( x^{3} +1\right)( x-1)\right]\\ & =\left( x^{2} +x+1\right)\left[ x^{5} -\left( x^{4} -x^{3} +x-1\right)\right]\\ & =\left( x^{2} +x+1\right)\left( x^{5} -x^{4} +x^{3} -x+1\right) \end{aligned} \end{equation}

伊另外有一題是 $\displaystyle x^{7} +x^{2} +1$，我想著會使利用類似伊頂面个技巧，不過略仔較複雜，因為愛增兩項、減兩項：

\begin{equation*} \begin{aligned} & \kern+0.4em \kern+0.4em \thinspace x^{7} +x^{2} +1\\ & =x^{7} +x^{6} +x^{5} -x^{5} +x^{2} -x^{6} +1\\ & =x^{5}\left( x^{2} +x+1\right) -x^{2}\left( x^{3} -1\right) -\left( x^{6} -1\right)\\ & =x^{5}\left( x^{2} +x+1\right) -x^{2}( x-1)\left( x^{2} +x+1\right) -\left( x^{3} +1\right)( x-1)\left( x^{2} +x+1\right)\\ & =\left( x^{2} +x+1\right)\left( x^{5} -x^{3} +x^{2} -x^{4} +x^{3} -x+1\right)\\ & =\left( x^{2} +x+1\right)\left( x^{5} -x^{4} +x^{2} -x+1\right) \end{aligned} \end{equation*}

我做出來了後去對照伊个解法，發現伊拄開始是增一項、減一項：

\begin{equation*} \begin{aligned} & \kern+0.4em \kern+0.4em \thinspace x^{7} +x^{2} +1\\ & =x^{7} -x^{4} +x^{4} +x^{2} +1\\ & =x^{4}\left( x^{3} -1\right) +\left( x^{4} +x^{2} +1\right)\\ & =x^{4}\left( x^{3} -1\right) +\left( x^{4} +2x^{2} +1-x^{2}\right)\\ & =x^{4}\left( x^{3} -1\right) +\left[\left( x^{2} +1\right)^{2} -x^{2}\right]\\ & =x^{4}( x-1)\left( x^{2} +x+1\right) +\left( x^{2} +x+1\right)\left( x^{2} -x+1\right)\\ & =\left( x^{2} +x+1\right)\left( x^{5} -x^{4} +x^{2} -x+1\right) \end{aligned} \end{equation*}

伊閣有一題，是佇上頭前迄條 $\displaystyle x^{7} +x^{5} +1=\left( x^{2} +x+1\right)\left( x^{5} -x^{4} +x^{3} -x+1\right)$ 个基礎頂面閣發揮：

\begin{equation*} \begin{aligned} & \kern+0.4em \kern+0.4em \thinspace x^{10} +x^{5} +1\\ & =x^{10} -x^{7} +x^{7} +x^{5} +1\\ & =x^{7}\left( x^{3} -1\right) +\left( x^{7} +x^{5} +1\right)\\ & =x^{7}( x-1)\left( x^{2} +x+1\right) +\left( x^{2} +x+1\right)\left( x^{5} -x^{4} +x^{3} -x+1\right)\\ & =\left( x^{2} +x+1\right)\left( x^{8} -x^{7} +x^{5} -x^{4} +x^{3} -x+1\right) \end{aligned} \end{equation*}