IGCSE Edexcel Further Pure Mathematics: Vocabulary & Terminology Memorisation Guide | IGCSE Edexcel 进阶数学：词汇术语速记指南

📚 IGCSE Edexcel Further Pure Mathematics: Vocabulary & Terminology Memorisation Guide | IGCSE Edexcel 进阶数学：词汇术语速记指南

Welcome to the IGCSE Edexcel Further Pure Mathematics vocabulary guide. Mastering key terminology is essential for understanding exam questions and communicating solutions clearly. This guide pairs each term with a concise definition and a memory tip to accelerate your revision.

欢迎使用 IGCSE Edexcel 进阶数学词汇速记指南。掌握关键术语对于理解考题和清晰表达解题过程至关重要。本指南为每个术语提供简洁定义和记忆小贴士，帮助你高效复习。

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1. Algebraic Expressions & Equations | 代数表达式与方程

A polynomial is an expression formed by adding terms of the form a₅xⁿ, where n is a non-negative integer. The degree is the highest exponent of x. The leading coefficient is the coefficient of the highest-degree term. For example, in P(x) = 4x⁵ − x³ + 2x, the degree is 5 and the leading coefficient is 4.

多项式是由形如 a₅xⁿ 的项相加而成的表达式，其中 n 是非负整数。次数是 x 的最高指数。首项系数是最高次项对应的系数。例如 P(x) = 4x⁵ − x³ + 2x 中，次数为 5，首项系数为 4。

The Remainder Theorem: When polynomial f(x) is divided by (x − a), the remainder is f(a). This provides a quick way to evaluate remainders without long division.

余数定理：多项式 f(x) 除以 (x − a) 时，余数为 f(a)。这提供了一种无需长除法就能快速求余数的方法。

The Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0. It is a special case of the Remainder Theorem and is widely used for factorising cubics and higher-degree polynomials.

因式定理：当且仅当 f(a)=0 时，(x − a) 是 f(x) 的因式。它是余数定理的特例，广泛用于对三次及更高次多项式进行因式分解。

The discriminant of a quadratic equation ax² + bx + c = 0 is Δ = b² − 4ac. It reveals the nature of roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots (two complex conjugate roots).

二次方程 ax² + bx + c = 0 的判别式为 Δ = b² − 4ac。它揭示根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重实根；Δ < 0 无实根（两个共轭复根）。

x = ( −b ± √(b² − 4ac) ) / 2a

The quadratic formula gives the solutions directly.

二次公式直接给出方程的解。

Completing the square transforms ax² + bx + c into a(x + p)² + q, revealing the vertex (−p, q) and the axis of symmetry.

配方法将 ax² + bx + c 化为 a(x + p)² + q，从而得到顶点 (−p, q) 和对称轴。

Simultaneous equations consist of two or more equations involving the same variables. They are solved by elimination or substitution. In Further Pure, we often solve one linear and one quadratic equation, yielding up to two pairs of solutions.

联立方程组由两个或更多含有相同变量的方程组成，可通过消元法或代入法求解。在进阶数学中常涉及一个线性和一个二次方程的联立，最多得到两对解。

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2. Sequences & Series | 数列与级数

An arithmetic sequence has a constant common difference d between consecutive terms. The n-th term is u₅ = a + (n − 1)d, and the sum of the first n terms is given by S₅ = n/2 [2a + (n−1)d] or S₅ = n/2 (a + l).

算术数列相邻项的差 d 为常数。第 n 项为 u₅ = a + (n − 1)d，前 n 项和为 S₅ = n/2 [2a + (n−1)d] 或 S₅ = n/2 (a + l)（l 为末项）。

S₅ = n/2 [2a + (n−1)d]

A geometric sequence has a constant common ratio r between consecutive terms. The n-th term is arⁿ⁻¹, and the sum of the first n terms is S₅ = a(1 − rⁿ) / (1 − r) for r ≠ 1.

几何数列相邻项的比值 r 为常数。第 n 项为 arⁿ⁻¹，前 n 项和为 S₅ = a(1 − rⁿ) / (1 − r)（r ≠ 1）。

When |r| < 1, the infinite sum converges: S∞ = a / (1 − r). This is a key concept for geometric series.

当 |r| < 1 时，无穷级数收敛：S∞ = a / (1 − r)。这是几何级数的重要概念。

Sigma notation Σ₉̀₁ⁿ a₉ compactly represents the sum of terms. The lower limit gives the starting index and the upper limit gives the ending index.

求和符号 Σ₉̀₁ⁿ a₉ 简洁地表示各项的和，下标给出起始索引，上标给出终止索引。

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3. Functions | 函数

A function maps each input x to exactly one output y = f(x). The domain is the set of all possible input values, and the range is the set of all output values produced.

函数将每个输入 x 映射到唯一的输出 y = f(x)。定义域是所有可能输入值的集合，值域是所有输出值的集合。

A one-to-one function gives a unique output for every distinct input, and it passes the horizontal line test. Many-to-one functions map different inputs to the same output and are not invertible without restricting the domain.

一一对应函数对每个不同输入给出唯一输出，并通过水平线检验。多对一函数将不同输入映射到相同输出，若非限制定义域则不可逆。

The composite function fg(x) means first apply g, then apply f to the result: f(g(x)). The order matters: gf(x) is generally different from fg(x).

复合函数 fg(x) 表示先应用 g，再对结果应用 f，即 f(g(x))。复合顺序很重要：gf(x) 通常与 fg(x) 不同。

The inverse function f⁻¹(x) reverses the effect of f, such that f(f⁻¹(x)) = x. It exists only if f is one-to-one, and its graph is the reflection of y = f(x) in the line y = x.

反函数 f⁻¹(x) 逆转 f 的作用，满足 f(f⁻¹(x)) = x。仅当 f 是一一对应时反函数才存在，其图像是 y = f(x) 关于直线 y = x 的对称像。

The modulus function |x| returns the non-negative distance of x from zero: |x| = x if x ≥ 0, and |x| = −x if x < 0. Its graph is V-shaped.

绝对值函数 |x| 返回 x 到零的非负距离：x ≥

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