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# 高校受験数学/解答と解説

## 大問1

### 解答

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

### 解説

#### 問1

${\displaystyle \displaystyle {\frac {\left({\sqrt {18}}-2{\sqrt {50}}+\displaystyle {\frac {6}{\sqrt {2}}}\right)^{2}}{5{\sqrt {2}}+{\sqrt {162}}-{\sqrt {200}}}}}$を計算せよ。

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

#### 問2

${\displaystyle (x^{2}+x+1)(x^{2}-x+1)(x^{4}-''x''^{2}+1)(x^{8}-x^{4}+1)}$を展開せよ。

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