📚 Cubic Equations with Related Roots | 三次方程的相关根
hose roots are 2α, 2β and 2γ, or 1/α, 1/β and 1/γ. The same technique extends to higher-degree equations, but the cubic is the most commonly tested case.
在AQA进阶数学中，相关根的三次方程要求我们构造一个新三次方程，其根是已知三次方程根的某种函数。例如，若α、β、γ满足 x³ − 6x² + 11x − 6 = 0，我们可能被要求找出以2α、2β、2γ或1/α、1/β、1/γ为根的三次方程。这一技巧同样适用于更高次方程，但三次方程是考查最频繁的情形。
1. Vieta's Formulas: The Engine of Related Roots | 韦达定理：相关根的引擎
cubic equation ax³ + bx² + cx + d = 0 with roots α, β and γ. The three symmetric sums are:
考虑一般三次方程 ax³ + bx² + cx + d = 0，其根为α、β、γ。三个对称和为：
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
These three quantities encode all the information needed to reconstruct the equation. No matter how the roots are transformed, if we can compute the three new symmetric sums, we can write down the new cubic.
这三个量包含了重建方程所需的全部信息。无论根如何变换，只要我们能算出三个新的对称和，就能写出新的三次方程。
For the monic form x³ + px² + qx + r = 0, the formulas simplify to:
对于首一项形式 x³ + px² + qx + r = 0，公式简化为：
α + β + γ = −p, αβ + βγ + γα = q, αβγ = −r
hose roots are 1, 2 and 3. This keeps the arithmetic transparent so you can focus on the method.
本文自始至终使用示例 x³ − 6x² + 11x − 6 = 0，其根为1、2、3。这使运算一目了然，便于你专注于方法本身。
2. Constructing the New Equation | 构造新方程
Suppose the new roots are α′, β′ and γ′, with symmetric sums:
设新根为α′、β′、γ′，其对称和为：
S₁ = α′ + β′ + γ′, S₂ = α′β′ + β′γ′ + γ′α′, S
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