题目

Problem

The equation $2x^2 + 5x + 7 = 0$ has roots $\alpha$ and $\beta$

Without solving the equation

(a) determine the exact value of $\alpha^3 + \beta^3$

(3)

(b) form a quadratic equation, with integer coefficients, which has roots

$\frac{\alpha^2}{\beta} \quad \text{and} \quad \frac{\beta^2}{\alpha}$

(5)

题目中文翻译

方程 $2x^2 + 5x + 7 = 0$ 的两根为 $\alpha$ 和 $\beta$ 。

不需求解该方程，

(a) 确定 $\alpha^3 + \beta^3$ 的精确值。

(b) 构造一个具有整数系数的二次方程，使其两根为 $\frac{\alpha^2}{\beta} \quad \text{和} \quad \frac{\beta^2}{\alpha}$

解答

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不直接求根，而是由韦达定理取得 $\alpha+\beta$ 与 $\alpha\beta$ 。再利用恒等式 $\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$ 求值。

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By Vieta’s formulae,

\begin{align*} \alpha+\beta=&\,-\frac52,\\ \alpha\beta=&\,\frac72. \end{align*}

Hence,

\begin{align*} \alpha^3+\beta^3 =&\,(\alpha+\beta)^3 -3\alpha\beta(\alpha+\beta)\\ =&\,\bigg(-\frac52\bigg)^3\\ &\,\hspace{2pt}-3\bigg(\frac72\bigg) \bigg(-\frac52\bigg)\\ =&\,-\frac{125}{8}+\frac{210}{8}\\ =&\,\boxed{\frac{85}{8}}. \end{align*}

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设新方程的两根为 $u=\alpha^2/\beta$ 与 $v=\beta^2/\alpha$ 。分别用 (a) 的结果和韦达定理求 $u+v$ 、 $uv$ ，再代入根和与根积所确定的二次方程 $x^2-(u+v)x+uv=0$ ，最后清除分母。

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Let

\begin{align*} u=&\,\frac{\alpha^2}{\beta},\\ v=&\,\frac{\beta^2}{\alpha}. \end{align*}

Using part (a), the sum of the new roots is

\begin{align*} u+v =&\,\frac{\alpha^2}{\beta}\\ &\,\hspace{2pt}+\frac{\beta^2}{\alpha}\\ =&\,\frac{\alpha^3+\beta^3}{\alpha\beta}\\ =&\,\frac{85}{28}. \end{align*}

Their product is

\begin{align*} uv =&\,\frac{\alpha^2\beta^2}{\alpha\beta}\\ =&\,\alpha\beta\\ =&\,\frac72. \end{align*}

Thus the required equation is initially

x^2-\frac{85}{28}x+\frac72=0.

Multiplying by $28$ gives the equation with integer coefficients,

\boxed{28x^2-85x+98=0}.