= ( 3 2 − 10 2 + 3 2 ) 2 5 2 + 9 2 − 10 2 = ( − 4 2 ) 2 4 2 = ( 4 2 ) 2 4 2 = 4 2 _ _ {\displaystyle {\begin{matrix}&=&\displaystyle {\frac {\left(3{\sqrt {2}}-10{\sqrt {2}}+3{\sqrt {2}}\right)^{2}}{5{\sqrt {2}}+9{\sqrt {2}}-10{\sqrt {2}}}}\\&=&\displaystyle {\frac {\left(-4{\sqrt {2}}\right)^{2}}{4{\sqrt {2}}}}\\&=&\displaystyle {\frac {\left(4{\sqrt {2}}\right)^{2}}{4{\sqrt {2}}}}\\&=&{\underline {\underline {4{\sqrt {2}}}}}\end{matrix}}}
= { ( x 2 + x + 1 ) ( x 2 − x + 1 ) } ( x 4 − x 2 + 1 ) ( x 8 − x 4 + 1 ) = { ( x 4 + 2 x 2 + 1 ) − x 2 } ( x 4 − x 2 + 1 ) ( x 8 − x 4 + 1 ) = ( x 4 + x 2 + 1 ) ( x 4 − x 2 + 1 ) ( x 8 − x 4 + 1 ) = ( x 8 + x 4 + 1 ) ( x 8 − x 4 + 1 ) = x 16 + x 8 + 1 _ _ {\displaystyle {\begin{matrix}&=&\left\{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)\right\}\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&\left\{\left(x^{4}+2x^{2}+1\right)-x^{2}\right\}\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&\left(x^{4}+x^{2}+1\right)\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&\left(x^{8}+x^{4}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&{\underline {\underline {x^{16}+x^{8}+1}}}\end{matrix}}}
左から掛けていくと、次々に和と差の積の公式が使える形が表れていくのがポイントである。
