Edexcel A-Level Mathematics Core Skills (PDF Joiner 4-236) | Edexcel A-Level 数学核心技能复习（PDF合并4-236）

📚 Edexcel A-Level Mathematics Core Skills (PDF Joiner 4-236) | Edexcel A-Level 数学核心技能复习（PDF合并4-236）

This revision note reviews key topics in Edexcel A-Level Mathematics pure content. It is designed for quick consolidation of the skills that frequently appear in AS and A2 papers, including algebra, trigonometry, calculus, numerical methods and vectors.

本复习笔记回顾 Edexcel A-Level 数学纯数部分的核心主题，旨在快速巩固 AS 与 A2 试卷中频繁出现的技能，包括代数、三角学、微积分、数值方法和向量。

1. Algebraic Expressions and the Factor Theorem | 代数表达式与因式定理

The factor theorem states that for a polynomial f(x), if f(a) = 0, then (x − a) is a factor of f(x). The remainder theorem gives f(a) as the remainder when f(x) is divided by (x − a).

因式定理指出，对于多项式 f(x)，若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式。余数定理给出用 (x − a) 除 f(x) 时的余数为 f(a)。

To factorise a cubic polynomial, first try small integer values such as ±1, ±2, ±3. After finding one linear factor, use long division or compare coefficients to reduce the cubic to a quadratic, then factorise the quadratic if possible.

分解三次多项式时，先尝试小整数值，如 ±1、±2、±3。找到一个一次因式后，使用长除法或比较系数法将三次式降为二次式，然后再尽可能分解二次式。

• Factor theorem: if f(a) = 0, then (x − a) is a factor | 因式定理：若 f(a) = 0，则 (x − a) 为因式
• Remainder theorem: remainder = f(a) when dividing by (x − a) | 余数定理：除以 (x − a) 的余数为 f(a)
• Always factorise fully unless the question states otherwise | 除非题目另有说明，否则要完全分解

2. Binomial Expansion | 二项式展开

The binomial expansion for (1 + x)ⁿ, where n is a positive integer, is finite. The coefficient of xʳ is given by ⁿCᵣ = n!/(r!(n − r)!). For (a + b)ⁿ, the general term is ⁿCᵣ aⁿ⁻ʳ bʳ.

当 n 为正整数时，(1 + x)ⁿ 的二项式展开是有限项。xʳ 的系数为 ⁿCᵣ = n!/(r!(n − r)!)。对于 (a + b)ⁿ，通项为 ⁿCᵣ aⁿ⁻ʳ bʳ。

When n is not a positive integer, the expansion of (1 + x)ⁿ is infinite and is valid only for |x| < 1. Edexcel questions often ask for the range of validity or for approximations with a given x value.

当 n 不是正整数时，(1 + x)ⁿ 的展开为无穷级数，并且仅在 |x| < 1 时有效。Edexcel 题目常要求给出有效范围，或用给定的 x 值进行近似计算。

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ

3. Trigonometry and Identities | 三角学与恒等式

You must know exact trigonometric values for 30°, 45° and 60° using sine, cosine and tangent. The key identities are tanθ = sinθ/cosθ and sin²θ + cos²θ ≡ 1.

必须熟记 30°、45° 和 60° 的正弦、余弦和正切精确值。核心恒等式为 tanθ = sinθ/cosθ 和 sin²θ + cos²θ ≡ 1。

To solve a trigonometric equation such as sin(2x + 30°) = 0.5, first find the principal value, then use the CAST diagram or graph to locate all solutions in the required interval. Remember to undo the transformation on the angle at the end.

解方程如 sin(2x + 30°) = 0.5 时，先求主值，再利用 CAST 图或图像在指定区间内找到所有解。最后记得反解出原角度 x。

• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 | 正弦 30° = 1/2，余弦 30° = √3/2，正切 30° = 1/√3
• sin 45° = cos 45° = 1/√2, tan 45° = 1 | 正弦 45° = 余弦 45° = 1/√2，正切 45° = 1
• sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3 | 正弦 60° = √3/2，余弦 60° = 1/2，正切 60° = √3

4. Exponentials and Logarithms | 指数与对数

The logarithm laws are logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, and logₐ(xᵏ) = k logₐx. The natural logarithm ln x has base e.

对数法则为 logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，以及 logₐ(xᵏ) = k logₐx。自然对数 ln x 的底为 e。

Exponential growth and decay models often use y = aeᵏᵗ. Taking natural logarithms reduces the model to a linear relationship: ln y = ln a + kt. This is useful when estimating constants from experimental data.

指数增长与衰减模型常使用 y = aeᵏᵗ。取自然对数可将模型化为线性关系：ln y = ln a + kt。这在根据实验数据估计常数时非常有用。

eˡⁿˣ = x and ln(eˣ) = x

5. Differentiation Techniques | 微分技巧

Standard derivatives include d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(eˣ) = eˣ, and d/dx(ln x) = 1/x.

标准导数包括 d/dx(xⁿ) = nxⁿ⁻¹，d/dx(sin x) = cos x，d/dx(cos x) = −sin x，d/dx(eˣ) = eˣ，以及 d/dx(ln x) = 1/x。

For products, use the product rule: if y = uv, then dy/dx = u’v + uv’. For quotients, use the quotient rule: if y = u/v, then dy/dx = (u’v − uv’)/v². For composite functions, apply the chain rule: dy/dx = dy/du × du/dx.

乘积使用乘法法则：若 y = uv，则 dy/dx = u’v + uv’。商的导数使用除法法则：若 y = u/v，则 dy/dx = (u’v − uv’)/v²。复合函数使用链式法则：dy/dx = dy/du × du/dx。