📚 GCSE AQA Further Maths Formula & Theorem Quick Reference Handbook | GCSE AQA 进阶数学：公式定理速查手册
Welcome to your ultimate quick-reference guide for GCSE AQA Further Mathematics. All essential formulas, theorems, and must-know facts are compiled here to support your revision and exam preparation.
欢迎使用 GCSE AQA 进阶数学终极速查手册。本指南汇编了所有核心公式、定理与必考要点，助你高效复习、从容应考。
1. Algebraic Basics | 代数基础
Quadratic Formula: For ax² + bx + c = 0, the solutions are:
x = (-b ± √b² - 4ac) / (2a)
二次公式：对于 ax² + bx + c = 0，解为：
x = (-b ± √b² - 4ac) / (2a)
The Discriminant: b² - 4ac. If Δ > 0 two distinct real roots, Δ = 0 one repeated root, Δ < 0 no real roots.
判别式：Δ = b² - 4ac。若 Δ > 0 有两个相异实根，Δ = 0 有一个重根，Δ < 0 无实根。
Sum and Product of Roots: For α and β as roots of ax² + bx + c = 0, α + β = -b/a, αβ = c/a.
根的和与积：若 α 和 β 是 ax² + bx + c = 0 的根，则 α + β = -b/a，αβ = c/a。
Completing the Square: x² + bx can be written as (x + b/2)² - (b/2)².
配方法：x² + bx 可写成 (x + b/2)² - (b/2)²。
Laws of Indices:
Multiplying: aᵐ × aⁿ = am+n
幂的乘法：aᵐ × aⁿ = am+n
Dividing: aᵐ ÷ aⁿ = am-n
幂的除法：aᵐ ÷ aⁿ = am-n
Power of a power: (aᵐ)ⁿ = amn
幂的幂：(aᵐ)ⁿ = amn
Negative exponent: a⁻ⁿ = 1/aⁿ
负指数：a⁻ⁿ = 1/aⁿ
Fractional exponent: a1/n = ⁿ√a, am/n = (ⁿ√a)ᵐ
分数指数：a1/n = ⁿ√a，am/n = (ⁿ√a)ᵐ
2. Polynomials & Factor Theorem | 多项式与因式定理
Remainder Theorem: When polynomial f(x) is divided by (x - a), the remainder is f(a).
余式定理：多项式 f(x) 除以 (x - a) 的余式为 f(a)。
Factor Theorem: (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。
Polynomial Division: Use long division or synthetic division to factorise cubics and quartics.
多项式除法：用长除法或综合除法分解三次或四次多项式。
Factorising cubic with no constant term: Factor out x first, then handle the remaining quadratic.
无常数项三次式的因式分解：先提取公因式 x，再处理剩余的二次式。
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