IGCSE CAIE Additional Mathematics: Formula & Theorem Quick Revision Handbook | IGCSE CAIE 进阶数学：公式定理速查手册

📚 IGCSE CAIE Additional Mathematics: Formula & Theorem Quick Revision Handbook | IGCSE CAIE 进阶数学：公式定理速查手册

This quick revision handbook summarises the essential formulae and theorems for the CAIE IGCSE Additional Mathematics (0606) syllabus. Use it to check definitions, memorise key results, and practise applying them under timed conditions.

本速查手册汇总 CAIE IGCSE 进阶数学（0606）考试所需的必备公式与定理，可用于核对定义、记忆关键结论，并在限时条件下练习应用。

1. Surds, Indices & Logarithms | 根式、指数与对数

For rational indices, the three core laws of indices are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ. Negative powers give reciprocals, and fractional powers give roots.

对于有理指数，三条核心指数律为 aᵐ × aⁿ = aᵐ⁺ⁿ、aᵐ ÷ aⁿ = aᵐ⁻ⁿ 以及 (aᵐ)ⁿ = aᵐⁿ。负指数表示倒数，分数指数表示根式。

a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a

To rationalise a denominator such as 1/(√a + √b), multiply the numerator and denominator by the conjugate √a − √b. This removes the square roots from the denominator.

要对分母如 1/(√a + √b) 进行有理化，可将分子与分母同乘共轭式 √a − √b，从而消去分母中的根号。

Logarithms reverse exponentiation. The statement logₐ x = y is equivalent to x = aʸ. The key log rules are logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, and logₐ xⁿ = n logₐ x.

对数是指数运算的逆运算。logₐ x = y 等价于 x = aʸ。关键对数法则为 logₐ(xy) = logₐ x + logₐ y、logₐ(x/y) = logₐ x − logₐ y、logₐ xⁿ = n logₐ x。

logₐ b = log c b ÷ log c a

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2. Factors of Polynomials & Cubic Equations | 多项式因式与三次方程

The remainder theorem states that when a polynomial f(x) is divided by (x − a), the remainder is f(a). If f(a) = 0, then (x − a) is a factor of f(x); this is the factor theorem.

余式定理指出，当多项式 f(x) 除以 (x − a) 时，余数为 f(a)。若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式；这就是因式定理。

f(x) = (x − a)Q(x) + R, R = f(a)

For a cubic equation, if one real root x = a is known, divide the cubic by (x − a) to obtain a quadratic. The remaining roots can then be found by factorising the quadratic or using the quadratic formula.

对于三次方程，若已知一个实根 x = a，可将三次多项式除以 (x − a) 得到二次式，再对二次式因式分解或使用求根公式求出剩余根。

Always check for common factors first, and look for a sum or difference of cubes such as a³ + b³ = (a + b)(a² − ab + b²) or a³ − b³ = (a − b)(a² + ab + b²).

解题时先检查是否有公因式，并注意立方和或立方差公式，如 a³ + b³ = (a + b)(a² − ab + b²) 或 a³ − b³ = (a − b)(a² + ab + b²)。

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3. Quadratic Functions & Inequalities | 二次函数与不等式

For the quadratic equation ax² + bx + c = 0, the roots are given by the quadratic formula. The discriminant Δ = b² − 4ac determines the nature of the roots: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.

对于二次方程 ax² + bx + c = 0，其根由求根公式给出。判别式 Δ = b² − 4ac 决定根的性质：若 Δ > 0 有两个不同实根，若 Δ = 0 有一个重实根，若 Δ < 0 则无实根。

x = [−b ± √(b² − 4ac)] ÷ 2a

If the roots are α and β, then the sum of roots is α + β = −b/a and the product of roots is αβ = c/a. Completing the square can be written as ax² + bx + c = a(x + b/2a)² + (c − b²/4a).

若两根为 α 与 β，则根的和为 α + β = −b/a，根的积为 αβ = c/a。配方法可写为 ax² + bx + c = a(x + b/2a)² + (c − b²/4a)。

For quadratic inequalities such as ax² + bx + c > 0, first solve the corresponding equation to find critical values, then test intervals on a sign diagram. Remember to reverse the inequality sign when multiplying or dividing by a negative number.

对于二次不等式如 ax² + bx + c > 0，先解对应方程求出临界值，再用符号表检验区间。注意当乘或除以负数时，不等号方向要改变。

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4. Binomial Expansion | 二项式展开

For a positive integer n, the binomial expansion of (a + b)ⁿ is the sum of terms nCr aⁿ⁻ʳ bʳ, where r runs from 0 to n. The binomial coefficient nCr is the number of ways to choose r items from n items.

对于正整数 n，(a + b)ⁿ 的二项式展开为 nCr aⁿ⁻ʳ bʳ 的各项之和，其中 r 从 0 取到 n。二项式系数 nCr 表示从 n 个对象中选出 r 个的方法数。

(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, nCr = n! ÷ [r!(n − r)!]

The (r + 1)th term in the expansion of (a + b)ⁿ is nCr aⁿ⁻ʳ bʳ. This term formula is useful when only one specific term is required rather than the entire expansion.

(a + b)ⁿ 展开式中的第 (r + 1) 项为 nCr aⁿ⁻ʳ bʳ。当只需要求某一特定项而不需要完整展开时，这个通项公式非常有用。

Pascal’s triangle gives the binomial coefficients quickly for small n: each entry is the sum of the two entries directly above it. The rows begin 1, then 1 1, 1 2 1, 1 3 3 1, and so on.

帕斯卡三角形可以快速给出较小 n 的二项式系数：每个数是其上方两个数之和。各行从 1 开始，依次为 1 1、1 2 1、1 3 3 1 等。

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5. Trigonometry & Circular Measure | 三角学与弧度制

Angles can be measured in degrees or radians. The conversion is π rad = 180°, so to convert from radians to degrees multiply by 180/π, and from degrees to radians multiply by π/180.

角可以用角度或弧度表示。换算关系为 π rad = 180°，因此弧度转角需乘以 180/π，角度转弧需乘以 π/180。

In circular measure, the arc length of a circle sector is s = rθ and the area of the sector is A = ½ r² θ, where r is the radius and θ is the angle in radians.

在弧度制中，扇形弧长为 s = rθ，扇形面积为 A = ½ r² θ，其中 r 为半径，θ 为弧度角。

s = rθ, A = ½ r² θ

Exact trigonometric values for 0°, 30°, 45°, 60° and 90° should be memorised. For example, sin 30° = ½, cos 60° = ½, tan 45° = 1, and sin 90° = 1.

应熟记 0°、30°、45°、60°、90° 的精确三角函数值。例如 sin 30° = ½，cos 60° = ½，tan 45° = 1，sin 90° = 1。

The sine rule and cosine rule are used for non-right-angled triangles. The sine rule is a/sin A = b/sin B = c/sin C, and the cosine rule is a² = b² + c² − 2bc cos A. The area of a triangle is ½ ab sin C.

正弦定理与余弦定理用于非直角三角形。正弦定理为 a/sin A = b/sin B = c/sin C，余弦定理为 a² = b² + c² − 2bc cos A。三角形面积为 ½ ab sin C。

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6. Trigonometric Identities & Equations | 三角恒等式与方程

The reciprocal and quotient identities are tan θ = sin θ / cos θ, cot θ = cos θ / sin θ, sec θ = 1/cos θ, and cosec θ = 1/sin θ.

倒数恒等式与商数恒等式为 tan θ = sin θ / cos θ、cot θ = cos θ / sin θ、sec θ = 1/cos θ、cosec θ = 1/sin θ。

The main Pythagorean identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 +

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