## 1. 問題の内容

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1. 問題の内容

2次方程式 $2x^2 - 3x + 8 = 0$ の2つの解を $\alpha, \beta$ とするとき、以下の式の値を求めよ。

(1) $\alpha^2\beta + \alpha\beta^2$

(2) $\alpha^2 + \beta^2$

(3) $\frac{\beta}{\alpha} + \frac{\alpha}{\beta}$

(4) $\alpha^3 + \beta^3$

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2. 解き方の手順

まず、解と係数の関係より、

$\alpha + \beta = -\frac{-3}{2} = \frac{3}{2}$

$\alpha\beta = \frac{8}{2} = 4$

これらを用いて、各問を解いていく。

(1) $\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta)$

$= 4 \times \frac{3}{2} = 6$

(2) $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$

$= (\frac{3}{2})^2 - 2 \times 4 = \frac{9}{4} - 8 = \frac{9 - 32}{4} = -\frac{23}{4}$

(3) $\frac{\beta}{\alpha} + \frac{\alpha}{\beta} = \frac{\beta^2 + \alpha^2}{\alpha\beta}$

$= \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{-\frac{23}{4}}{4} = -\frac{23}{16}$

(4) $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$

$= (\frac{3}{2})^3 - 3 \times 4 \times \frac{3}{2} = \frac{27}{8} - 18 = \frac{27 - 144}{8} = -\frac{117}{8}$

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3. 最終的な答え

(1) $\alpha^2\beta + \alpha\beta^2 = 6$

(2) $\alpha^2 + \beta^2 = -\frac{23}{4}$

(3) $\frac{\beta}{\alpha} + \frac{\alpha}{\beta} = -\frac{23}{16}$

(4) $\alpha^3 + \beta^3 = -\frac{117}{8}$