三次方程 $2x^3+5x^2-6=0$ 的根为 $\alpha$、$\beta$、$\gamma$。

(a) 求一个以 $\dfrac1{\alpha^3}$、$\dfrac1{\beta^3}$、$\dfrac1{\gamma^3}$ 为根的三次方程。

(b) 求 $\dfrac1{\alpha^6}+\dfrac1{\beta^6}+\dfrac1{\gamma^6}$ 的值。

(c) 另求 $\dfrac1{\alpha^9}+\dfrac1{\beta^9}+\dfrac1{\gamma^9}$ 的值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

设新根为 $y=x^{-3}$。从原方程隔出 $x^2$，立方后用 $x^3=1/y$ 消去原变量。

The original roots are nonzero because the constant term is $-6$. Put $y=1/x^3$. From

we have $5x^2=6-2/y$. Cubing gives

Since $x^6=1/y^2$, multiplying by $y^3$ yields

Thus a required cubic equation is

把(a)的新根记为 $p,q,r$。原来的六次倒数和就是 $p^2+q^2+r^2$，由根与系数关系计算。

Let $p=\alpha^{-3}$, $q=\beta^{-3}$ and $r=\gamma^{-3}$. From part (a),

Therefore

每个新根都满足(a)的三次方程；把三条等式相加，再代入(b)的平方和。

Each of $p,q,r$ satisfies $216y^3=216y^2+53y+8$. Adding these three equations,

Hence