3つの解の対称式の値（A）

解と係数の関係より \[ \alpha+\beta+\gamma=2,\qquad \alpha\beta+\beta\gamma+\gamma\alpha=3,\qquad \alpha\beta\gamma=4 \] (1) $\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)=4-6=-2$ (2) $\alpha^3+\beta^3+\gamma^3-3\alpha\beta\gamma=(\alpha+\beta+\gamma)(\alpha^2+\beta^2+\gamma^2-\alpha\beta-\beta\gamma-\gamma\alpha)$ であるから \[ \alpha^3+\beta^3+\gamma^3=2\cdot(-2-3)+3\cdot 4=2 \] (3) $(\alpha+1)(\beta+1)(\gamma+1)=\alpha\beta\gamma+(\alpha\beta+\beta\gamma+\gamma\alpha)+(\alpha+\beta+\gamma)+1=4+3+2+1=10$ (4) $\dfrac{1}{\alpha}+\dfrac{1}{\beta}+\dfrac{1}{\gamma}=\dfrac{\beta\gamma+\gamma\alpha+\alpha\beta}{\alpha\beta\gamma}=\dfrac{3}{4}$