CIE IGCSE Additional Mathematics Formula & Theorem Quick Reference Handbook | CIE IGCSE 进阶数学公式定理速查手册

📚 CIE IGCSE Additional Mathematics Formula & Theorem Quick Reference Handbook | CIE IGCSE 进阶数学公式定理速查手册

This handbook provides a concise collection of all essential formulas and theorems required for the CIE IGCSE Additional Mathematics (0606) syllabus. Use it as a quick reference while revising algebra, functions, trigonometry, calculus, and vectors. Every formula is accompanied by a brief explanation in both English and Chinese to reinforce understanding.

本手册紧扣 CIE IGCSE Additional Mathematics (0606) 考纲，汇总了所有高频公式与核心定理，方便同学们在复习代数、函数、三角、微积分和向量时快速查阅。每个公式都配有中英双语解释，帮助大家巩固理解。

1. Algebra and Surds | 代数与根式

Law of indices – product rule: when multiplying terms with the same base, add the exponents.

aᵐ × aⁿ = aᵐ⁺ⁿ

指数乘法法则：同底数幂相乘，指数相加。

Law of indices – power of a power: multiply the exponents.

(aᵐ)ⁿ = aᵐⁿ

指数幂的幂法则：括号外乘方时，指数相乘。

Law of indices – negative and zero exponents: a⁻ⁿ = 1/aⁿ, and a⁰ = 1 (a ≠ 0).

a⁻ⁿ = 1/aⁿ, a⁰ = 1

负指数与零指数法则：a⁻ⁿ = 1/aⁿ，任何非零数的零次幂等于 1。

Fractional exponent represents a root: a^(1/n) = ⁿ√a, and more generally a^(m/n) = (ⁿ√a)ᵐ.

a^(1/n) = ⁿ√a, a^(m/n) = (ⁿ√a)ᵐ

分数指数表示根式：a^(1/n) = ⁿ√a，推广可得 a^(m/n) = (ⁿ√a)ᵐ。

Key factorisation identities: difference of two squares and perfect square expansions.

a² − b² = (a − b)(a + b), a² ± 2ab + b² = (a ± b)²

乘法公式与因式分解：平方差 a² − b² = (a − b)(a + b)；完全平方 a² ± 2ab + b² = (a ± b)²。

Surd simplification: √(ab) = √a × √b (a, b > 0) and rationalising the denominator: 1/√a = √a/a.

√(ab) = √a √b, 1/√a = √a / a

根式化简：√(ab) = √a × √b（a, b 为正数）；分母有理化：1/√a = √a / a。

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2. Quadratic Equations and Functions | 二次方程与函数

The standard quadratic equation: ax² + bx + c = 0. Its discriminant Δ = b² − 4ac determines the nature of roots.

Δ = b² − 4ac

标准二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有重根；Δ < 0 无实根。

The quadratic formula gives the solutions for any quadratic equation.

x = (−b ± √(b² − 4ac)) / (2a)

求根公式适用于所有二次方程，可直接代入系数求得根。

Completing the square rewrites ax² + bx + c in vertex form a(x − h)² + k, where (h, k) is the vertex of the parabola.

a(x − h)² + k

完全平方配方将二次函数化为顶点式 a(x − h)² + k，顶点为 (h, k)，对称轴为 x = h。

Sum and product of roots: for ax² + bx + c = 0, sum of roots α + β = −b/a, product αβ = c/a.

α + β = −b/a, αβ = c/a

韦达定理：二次方程两根之和 α + β = −b/a，两根之积 αβ = c/a，常用于不解方程求对称式的值。

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3. Inequalities | 不等式

Solving linear inequalities: treat them like equations, but reverse the inequality sign when multiplying or dividing by a negative number.

If a < b and c < 0, then ac > bc

解线性不等式与解方程类似，但乘以或除以负数时，不等号方向要改变。

Quadratic inequalities (ax² + bx + c > 0) are solved by considering the sign of the quadratic expression using a sketch graph or sign table. First find the roots, then determine the intervals where the inequality holds.

(x − p)(x − q) > 0 ⇒ x < p or x > q (if p < q)

二次不等式的解集可借助抛物线图像或符号表：先求出两根，再根据开口方向写出满足不等式的 x 区间。

Absolute value inequality |x − k| < d represents the distance from x to k is less than d; it is equivalent to −d < x − k < d.

|x − k| < d ⇔ −d < x − k < d

绝对值不等式 |x − k| < d 表示 x 到 k 的距离小于 d，等价于 −d < x − k < d，直接解出 x 的范围。

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4. Polynomials and Binomial Theorem | 多项式与二项式定理

Remainder Theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a).

Remainder = f(a)

余数定理：多项式 f(x) 除以 (x − a) 的余数等于 f(a)，可用于快速求余数。

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

f(a) = 0 ⇔ (x − a) is a factor

因式定理：若 f(a) = 0，则 (x − a) 是 f(x) 的因式，反之亦然，常用于因式分解高次多项式。

Binomial expansion for positive integer n: (a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹b + … + ⁿCᵣ aⁿ⁻ʳ bʳ + … + ⁿCₙ bⁿ, where ⁿCᵣ = n!/(r!(n − r)!).

(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ

二项式定理（n 为正整数）：(a + b)ⁿ 的通项公式为 Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ，组合数 ⁿCᵣ 可从帕斯卡三角形或公式求得。

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5. Functions | 函数

The domain of a function f(x) is the set of all possible input values (x), and the range is the set of all possible output values (f(x)). For example, f(x) = √(x − 2) has domain x ≥ 2.

Domain: x ≥ 2, Range: f(x) ≥ 0

函数的定义域是所有可能输入 x

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