Year 12 CIE Further Mathematics: Quick-Reference Formula & Theorem Handbook | CIE 进阶数学公式定理速查手册

📚 Year 12 CIE Further Mathematics: Quick-Reference Formula & Theorem Handbook | CIE 进阶数学公式定理速查手册

This concise handbook covers the essential formulas and theorems from the Year 12 CIE Further Mathematics course (AS Level, 9231). It is designed for quick revision of Further Pure Mathematics 1 topics, including polynomial roots, series, matrices, complex numbers, polar coordinates, induction, vectors, and numerical methods.

本手册涵盖 CIE 进阶数学（AS Level，9231）核心公式与定理，便于快速复习进阶纯数学 1 的知识点，包括多项式根、级数、矩阵、复数、极坐标、归纳法、向量与数值方法。

1. Roots of Polynomial Equations | 多项式方程的根与系数

For a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0 with roots α₁, α₂, …, αₙ, the sum of roots Σαᵢ = −aₙ₋₁ / aₙ, the sum of products of two distinct roots Σαᵢαⱼ = aₙ₋₂ / aₙ, the sum of triple products Σαᵢαⱼαₖ = −aₙ₋₃ / aₙ, and so on with alternating signs, ending with the product of all roots = (−1)ⁿ a₀ / aₙ.

对于多项式 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0，其根记为 α₁, α₂, …, αₙ，所有根之和 Σα = −aₙ₋₁ / aₙ，所有两根乘积之和 Σαβ = aₙ₋₂ / aₙ，所有三根乘积之和 Σαβγ = −aₙ₋₃ / aₙ，依此类推，符号交替，最后所有根的乘积为 (−1)ⁿ a₀ / aₙ。

For a quadratic ax² + bx + c = 0: α + β = −b/a, αβ = c/a.

二次方程 ax² + bx + c = 0：α + β = −b/a，αβ = c/a。

For a cubic ax³ + bx² + cx + d = 0: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

三次方程 ax³ + bx² + cx + d = 0：α + β + γ = −b/a，αβ + βγ + γα = c/a，αβγ = −d/a。

For a quartic ax⁴ + bx³ + cx² + dx + e = 0: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.

四次方程 ax⁴ + bx³ + cx² + dx + e = 0：Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。

A useful identity: Σα² = (Σα)² − 2Σαβ. For a cubic, this becomes (‒b/a)² − 2(c/a).

常用恒等式：Σα² = (Σα)² − 2Σαβ。对于三次方程，即 (‒b/a)² − 2(c/a)。

2. Summation of Series | 级数求和

The standard results for sums of powers of the first n natural numbers are:

前 n 个自然数的幂次和标准结果如下：

Σᵢ₌₁ⁿ r = ½ n(n+1)    |    Σᵢ₌₁ⁿ r² = ⅙ n(n+1)(2n+1)    |    Σᵢ₌₁ⁿ r³ = ¼ n²(n+1)²

Also, the sum of a constant c over n terms is simply nc. These formulas can be combined linearly: Σ (ar² + br + c) = a Σr² + b Σr + c Σ1.

此外，常数 c 的 n 项之和为 nc。这些公式可以线性组合：Σ (ar² + br + c) = a Σr² + b Σr + c Σ1。

3. Matrices: Operations and Determinants | 矩阵运算与行列式

For a 2×2 matrix

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