IGCSE Edexcel Further Pure Mathematics: Formula & Theorems Quick Reference Guide | IGCSE Edexcel 进阶数学：公式定理速查手册

📚 IGCSE Edexcel Further Pure Mathematics: Formula & Theorems Quick Reference Guide | IGCSE Edexcel 进阶数学：公式定理速查手册

This quick reference handbook compiles all essential formulas and theorems required for the Edexcel IGCSE Further Pure Mathematics syllabus. It is designed for rapid revision and easy recall during your exam preparation. Each section groups key results with concise explanations, ensuring you can find the right tool for any problem.

本速查手册汇编了 Edexcel IGCSE 进阶数学大纲所需的全部核心公式与定理，旨在帮助你在备考中快速回顾与记忆。每个部分按主题组织结果，配以简明解释，确保你能针对任何问题找到正确工具。

1. Polynomials and Binomial Expansion | 多项式与二项展开式

For a polynomial f(x), the Remainder Theorem states: when f(x) is divided by (x – a), the remainder is f(a).

余数定理：当多项式 f(x) 除以 (x – a) 时，余数为 f(a)。

The Factor Theorem follows: (x – a) is a factor of f(x) if and only if f(a) = 0.

因式定理：(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。

For a quadratic equation ax² + bx + c = 0 with roots α and β:

对于二次方程 ax² + bx + c = 0，根为 α 和 β：

α + β = –b/a, αβ = c/a

For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ:

对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ：

α + β + γ = –b/a, αβ + βγ + γα = c/a, αβγ = –d/a

The binomial expansion for rational index n, valid for |x| < 1:

有理指数 n 的二项展开式，|x| < 1 时成立：

(1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + n(n – 1)(n – 2)/3! x³ + …

For positive integer n, the expansion of (a + b)ⁿ uses binomial coefficients nCr:

对于正整数 n，(a + b)ⁿ 的展开式使用二项式系数 nCr：

(a + b)ⁿ = Σ (from r=0 to n) nCr a^{n – r} b^r

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2. Inequalities | 不等式

rmine the sign of the quadratic expression by sketching the graph or using a sign diagram.

解二次不等式 ax² + bx + c > 0，先求出方程 ax² + bx + c = 0 的根，然后通过画图或符号表确定二次式的正负。

For rational inequalities, bring all terms to one side so you have a single fraction compared to zero, then identify critical values where numerator or denominator equals zero, and use a sign table.

对于分式不等式，将所有项移到一边，化为单个分式与 0 比较，找出分子或分母为零的临界值，再用符号表。

Modulus inequalities can be rewritten without the modulus sign:

绝对值不等式可去掉绝对值符号改写：

|f(x)| < a ⇔ –a < f(x) < a

|f(x)| > a ⇔ f(x) < –a or f(x) > a

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3. Complex Numbers | 复数

A complex number is written z = a + bi, where i² = –1. Its complex conjugate is z* = a – bi, and its modulus is |z| = √a² + b². The argument is θ = arctan(b/a), taking care of the quadrant.

复数记作 z = a + bi，其中 i² = –1。其共轭复数为 z* = a – bi，模为 |z| = √a² + b²。辐角为 θ = arctan(b/a)，并需根据象限确定。

The polar form uses modulus and argument:

极坐标形式使用模和辐角：

z = r (cos θ + i sin θ) = r e^iθ

De Moivre's theorem is essential for powers:

棣莫弗定理对于求幂至关重要：

(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

To find the n-th roots of a complex number:

求复数的 n 次方根：

z^1/n = r^1/n [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], k = 0, 1, …, n–1

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4. Matrices | 矩阵

For a 2×2 matrix A = [a b; c d], the determinant is det(A) = ad – bc. If det(A) ≠ 0, the inverse exists:

对于 2×2 矩阵 A = [a b; c d]，行列式为 det(A) = ad – bc。若 det(A) ≠ 0，则逆矩阵存在：

A⁻¹ = (1/det(A)) [d –b; –c a]

has no inverse. The linear system AX = B can be solved by X = A⁻¹ B when A is invertible.

若 det(A) = 0，矩阵为奇异矩阵，无逆矩阵。当 A 可逆时，线性方程组 AX = B 的解为 X = A⁻¹ B。

2×2 transformation matrices represent linear transformations in the plane:

2×2 变换矩阵表示平面上的线性变换：

Reflection in x-axis: [1 0; 0 -1]. Reflection in y-axis: [-1 0; 0 1]. Rotation through angle θ anticlockwise: [cosθ –sinθ; sinθ cosθ]. Enlargement scale factor k: [k 0; 0 k]. Shear parallel to x-axis of factor a: [1 a; 0 1].

关于 x 轴对称：[1 0; 0 -1]。关于 y 轴对称：[-1 0; 0 1]。逆时针旋转 θ 角：[cosθ –sinθ; sinθ cosθ]。放大比例因子 k：[k 0; 0 k]。平行于 x 轴且因子为 a 的剪切：[1 a; 0 1]。

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5. Series and Summation | 级数与求和

arithmetic progression has first term a, common difference d. The n-th term is uₙ = a + (n – 1)d, and the sum of the first n terms is Sₙ = n/2 (2a + (n – 1)d) = n/2 (a + l), where l is the last term.

等差数列，首项为 a，公差为 d。第 n 项为 uₙ = a + (n – 1)d，前 n 项和为 Sₙ = n/2 (2a + (n – 1)d) = n/2 (a + l)，其中 l 为末项。

A geometric progression has first term a, common ratio r. The n-th term is uₙ = a r^n–1. Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. For |r| < 1, the sum to infinity is S∞ = a/(1 – r).

等比数列，首项为 a，公比为 r。第 n 项为 uₙ = a r^n–1。前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r)（r ≠ 1）。当 |r| < 1 时，无穷项和为 S∞ = a/(1 – r)。

Standard summations of powers of integers are used in series work:

整数幂的标准求和用于级数问题：

∑ (from r=1 to n) r = n(n+1)/2

∑ r² = n(n+1)(2n+1)/6, ∑ r³ = n²(n+1)²/4

The method of differences works when the general term can be expressed as f(r) – f(r+1), leading to telescoping cancellation.

裂项法在通项可表示为 f(r) – f(r+1) 时使用，通过相互抵消求和。

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6. Differentiation | 微分

Standard derivatives you must know include:

必须掌握的标准导数包括：

d/dx (xⁿ) = nxⁿ⁻¹, d/dx (e^x) = e^x, d/dx (ln x) = 1/x

d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (tan x) = sec² x

The chain rule, product rule, and quotient rule are central:

链式法则、乘法法则和除法法则是核心：

dy/dx = dy/du × du/dx, d/dx (uv) = u'v + uv', d/dx (u/v) = (u'v – uv')/v²

t) / (dx/dt). Implicit differentiation treats y as a function of x, differentiating both sides of an equation and using the chain rule for terms involving y.

对于参数方程 x = f(t), y = g(t)，导数为 dy/dx = (dy/dt) / (dx/dt)。隐函数微分将 y 视为 x 的函数，方程两边同时对 x 求导，含 y 的项使用链式法则。

Second derivatives are obtained by differentiating the first derivative, written d²y/dx², and indicate concavity.

二阶导数由一阶导数再次微分得到，记作 d²y/dx²，用于判断凹凸性。

Hyperbolic derivatives:

双曲函数的导数：

d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x

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7. Integration | 积分

Basic integrals are the reverse of differentiation:

基本积分是微分的逆运算：

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1), ∫ 1/x dx = ln|x| + C

∫ e^x dx = e^x + C, ∫ sin x dx = –cos x + C, ∫ cos x dx = sin x + C

by substitution: replace part of the integrand with a new variable u, so that ∫ f(g(x)) g'(x) dx becomes ∫ f(u) du. For definite integrals, change the limits accordingly.

换元积分法：将被积函数的一部分替换为新变量 u，使 ∫ f(g

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