高校受験数学/解答と解説

大問1

解答

1. $4{\sqrt {2}}$
2. $x^{16}+x^{8}+1$
3. $\left(x^{2}+1\right)\left(x+1\right)\left(x-1\right)$
4. a ≦ -7
5. a < 10
6. b = 4
7. 小問の解答
   1. $a={\sqrt {2}}$
   2. $x=2\pm {\sqrt {3}}$
   3. t = 2.75, 3

解説

問1

${\begin{matrix}&=&\displaystyle {\frac {\left(3{\sqrt {2}}-10{\sqrt {2}}+3{\sqrt {2}}\right)^{2}}{5{\sqrt {2}}+9{\sqrt {2}}-10{\sqrt {2}}}}\\&=&\displaystyle {\frac {\left(-4{\sqrt {2}}\right)^{2}}{4{\sqrt {2}}}}\\&=&\displaystyle {\frac {\left(4{\sqrt {2}}\right)^{2}}{4{\sqrt {2}}}}\\&=&{\underline {\underline {4{\sqrt {2}}}}}\end{matrix}}$

問2

${\begin{matrix}&=&\left\{\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)\right\}\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&\left\{\left(x^{4}+2x^{2}+1\right)-x^{2}\right\}\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&\left(x^{4}+x^{2}+1\right)\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&\left(x^{8}+x^{4}+1\right)\left(x^{8}-x^{4}+1\right)\\&=&{\underline {\underline {x^{16}+x^{8}+1}}}\end{matrix}}$

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。