📚 GCSE Eduqas Further Maths: Glossary Quick-Memorisation Guide | GCSE Eduqas 进阶数学：词汇术语速记指南
This guide provides a rapid revision of essential terminology for the GCSE Eduqas Further Mathematics specification. Mastering these key terms will help you tackle exam questions with confidence and precision. Each entry pairs a clear English definition with a Chinese translation to support bilingual learners.
本指南为 GCSE Eduqas 进阶数学考试提供核心术语的快速复习。掌握这些关键术语将帮助你有信心且准确地解答考题。每个条目都配有清晰的英文定义和中文翻译，以支持双语学习者。
1. Algebraic Expressions & Polynomials | 代数式与多项式
Polynomial: An expression constructed from variables and constants using addition, subtraction, multiplication, and non-negative integer exponents. Example: 3x⁴ - 2x² + 7.
多项式：由变量和常数通过加法、减法、乘法及非负整数指数构成的表达式。例如：3x⁴ - 2x² + 7。
Degree of a polynomial: The highest power of the variable in a polynomial. In 5x³ + 2x - 1, the degree is 3.
多项式的次数：多项式中变量的最高指数。在 5x³ + 2x - 1 中，次数为 3。
Factor Theorem: if f(a) = 0, then (x - a) is a factor. Conversely, if (x - a) is a factor, then f(a) = 0.
因式定理：对于多项式 f(x)，若 f(a) = 0，则 (x - a) 是一个因式。反之，若 (x - a) 是因式，则 f(a) = 0。
Remainder Theorem: When a polynomial f(x) is divided by (x - a), the remainder is f(a).
余数定理：多项式 f(x) 除以 (x - a) 时，余数为 f(a)。
2. Functions & Graphs | 函数与图像
Function: A rule that maps each input value to exactly one output value. Notation: f(x) = x² + 3.
函数：将每一个输入值唯一映射到一个输出值的规则。记作：f(x) = x² + 3。
Domain: The set of all possible input values (x-values) for which a function is defined.
定义域：函数有定义的所有可能输入值（x 值）的集合。
Range: The set of all possible output values (y-values) produced by a function.
值域：函数产生的所有可能输出值（y 值）的集合。
Inverse function: reverses the effect of the original function, denoted f⁻¹(x). It satisfies f(f⁻¹(x)) = x.
反函数：反转原函数作用的函数，记作 f⁻¹(x)。它满足 f(f⁻¹(x)) = x。
Composite function: A function formed by applying one function to the result of another, written as f(g(x)) or (f ∘ g)(x).
复合函数：将一个函数应用于另一个函数结果所构成的函数，写作 f(g(x)) 或 (f ∘ g)(x)。
3. Coordinate Geometry & Circles | 坐标几何与圆
Gradient of a straight line: The measure of steepness, calculated as rise / run or (y₂ - y₁) / (x₂ - x₁).
直线的斜率：陡峭程度的度量，计算为垂直变化 / 水平变化，即 (y₂ - y₁) / (x₂ - x₁)。
Equation of a circle (centre origin): A circle centred at (0,0) with radius r has equation x² + y² = r².
圆的方程（圆心在原点）：以 (0,0) 为圆心、r 为半径的圆方程为 x² + y² = r²。
General circle equation: For a circle with centre (a, b) and radius r, the equation is (x - a)² + (y - b)² = r².
一般圆方程：圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。
Tangent to a circle: A straight line that touches the circle at exactly one point. The radius at the point of contact is perpendicular to the tangent.
圆的切线：与圆恰好在一个点接触的直线。接触点处的半径与切线垂直。
4. Trigonometry & Radians | 三角学与弧度
Radian: ended at the centre of a circle by an arc equal in length to the radius. π radians = 180°.
弧度：角度度量单位，一弧度是弧长等于半径的圆弧所对的圆心角。π 弧度 = 180°。
Sine, cosine, tangent: ngle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
正弦、余弦、正切：在直角三角形中，对于角 θ：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Trigonometric identities: Key identities include sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.
三角恒等式：关键恒等式包括 sin²θ + cos²θ = 1 以及 tan θ = sin θ / cos θ。
CAST diagram: A mnemonic to remember which trigonometric ratios are positive in each quadrant: All, Sin, Tan, Cos in quadrants 1 to 4.
CAST 图：用于记忆各象限中正弦、余弦、正切符号的助记图：第一象限 All（全正），第二象限 Sin，第三象限 Tan，第四象限 Cos。
5. Vectors | 向量
Vector: A quantity that has both magnitude and direction, often represented as a column vector (x y) or as xi + yj.
向量：既有大小又有方向的量，通常表示为列向量 (x y) 或 xi + yj。
Magnitude of a vector: The length of the vector. For vector v = a i + b j, magnitude |v| = √a² + b².
向量的模：向量的长度。对于向量 v = a i + b j，模 |v| = √a² + b²。
Unit vector: A vector with magnitude 1. The unit vector in the direction of v is v / |v|.
单位向量：模为 1 的向量。v 方向上的单位向量为 v / |v|。
Dot product (scalar product): For vectors a and b, a·b = |a||b| cos θ, also computed as a₁b₁ + a₂b₂ in two dimensions.
点积（标量积）：对于向量 a 和 b，a·b = |a||b| cos θ，在二维中也可计算为 a₁b₁ + a₂b₂。
6. Matrices & Transformations | 矩阵与变换
Matrix: ngular array of numbers arranged in rows and columns. The order is m × n (rows × columns).
矩阵：按行和列排列的数字矩形阵列。阶数为 m × n（行数 × 列数）。
Matrix multiplication: The product AB is defined only if the number of columns in A equals the number of rows in B. The element in row i, column j is the dot product of row i of A with column j of B.
矩阵乘法：仅当 A 的列数等于 B 的行数时，乘积 AB 才有定义。第 i 行第 j 列的元素是 A 的第 i 行与 B 的第 j 列的点积。
Determinant of a 2×2 matrix: , b], [c, d]], det(M) = ad - bc. If det(M) = 0, the matrix is singular and has no inverse.
2×2 矩阵的行列式：对于 M = [[a, b], [c, d]]，det(M) = ad - bc。若 det(M) = 0，该矩阵是奇异矩阵，没有逆矩阵。
Transformation matrices: tric transformations: rotation, reflection, enlargement, shear. For example, [[0, -1], [1, 0]] represents a 90° rotation anticlockwise.
变换矩阵：常见的 2×2 矩阵表示几何变换：旋转、反射、放大、剪切。例如 [[0, -1], [1, 0]] 表示绕原点逆时针旋转 90°。
7. Differentiation | 微分
Derivative: The rate of change of a function. For f(x), the derivative f'(x) or dy/dx gives the gradient of the tangent to the curve at any point.
导数：函数的变化率。对于 f(x)，导数 f'(x) 或 dy/dx 表示曲线上任意一点切线的斜率。
Power rule for differentiation: If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹.
幂函数求导法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。
Stationary point: curve where dy/dx = 0. It can be a local maximum, local minimum, or a point of inflection.
驻点：曲线上 dy/dx = 0 的点。它可以是局部最大值、局部最小值或拐点。
Second derivative: The derivative of the derivative, denoted f''(x) or d²y/dx². It helps determine the nature of stationary points (concavity).
二阶导数：导数的导数，记作 f''(x) 或 d²y/dx²。它帮助确定驻点的性质（凹凸性）。
8. Integration | 积分
Indefinite integral: The reverse process of differentiation. ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration.
不定积分：微分的逆运算。∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)，C 为积分常数。
Power rule for integration: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, provided n ≠ -1.
幂函数积分法则：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，前提是 n ≠ -1。
Definite integral: d b, ∫ₐᵇ f(x) dx, represents the signed area under the curve y = f(x) from x = a to x = b.
定积分：在上下限 a 和 b 之间的积分 ∫ₐᵇ f(x) dx，表示曲线 y = f(x) 从 x = a 到 x = b 下方的带符号面积。
Area between curves: ves y = f(x) and y = g(x) from a to b is given by ∫ₐᵇ (f(x) - g(x)) dx, where f(x) ≥ g(x).
曲线间的面积：从 a 到 b 由两条曲线 y = f(x) 和 y = g(x) 围成的面积等于 ∫ₐᵇ (f(x) - g(x)) dx，其中 f(x) ≥ g(x)。
9. Sequences & Binomial Expansion | 数列与二项展开
Arithmetic sequence: onsecutive terms is constant, known as the common difference d. nth term: uₙ = a + (n-1)d.
等差数列：相邻两项之差为常数的数列，该常数称为公差 d。第 n 项：uₙ = a + (n-1)d。
Geometric sequence: rm is found by multiplying the previous term by a constant ratio r. nth term: uₙ = a rⁿ⁻¹.
等比数列：每一项由前一项乘以常数比 r 得到的数列。第 n 项：uₙ = a rⁿ⁻¹。
Binomial expansion: expands into a sum of terms involving binomial coefficients. (1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + ...
二项展开：对于正整数 n，(a + b)ⁿ 展开为涉及二项式系数的各项之和。(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + …
Binomial coefficient: Often written as ⁿCᵣ or (n r), calculated as n! / [r!(n-r)!]. It gives the coefficient of the xʳ term in the expansion of (1 + x)ⁿ.
二项式系数：通常写作 ⁿCᵣ 或 (n r)，计算为 n! / [r!(n-r)!]。它给出 (1 + x)ⁿ 展开式中 xʳ 项的系数。
10. Inequalities & Linear Programming | 不等式与线性规划
Linear inequality: inear expressions, such as 2x + 3y ≤ 6. The solution set is typically a region on a graph.
线性不等式：包含线性表达式的不等式，如 2x + 3y ≤ 6。解集通常是图形上的一个区域。
Quadratic inequality: An inequality such as x² - 5x + 6 > 0, solved by finding critical values and testing intervals or using a sign diagram.
二次不等式：如 x² - 5x + 6 > 0 的不等式，可通过求临界值并检验区间或使用符号图求解。
Linear programming: A method to find the maximum or minimum value of an objective function subject to linear constraints. The optimal solution lies at a vertex of the feasible region.
线性规划：在给定线性约束条件下，求目标函数最大值或最小值的方法。最优解位于可行区域的顶点处。
Feasible region: The set of all points that satisfy all the constraint inequalities simultaneously. It is often a convex polygon.
可行域：同时满足所有约束不等式的点集，通常是一个凸多边形。
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