📚 Year 10 OCR Further Maths: Quick Guide to Key Terminology | Year 10 OCR 进阶数学：词汇术语速记指南
Mastering the language of advanced mathematics is essential for success in OCR Year 10 Further Maths. This quick guide breaks down key terms into bite-size English and Chinese explanations, helping you build confidence and avoid common misunderstandings. Whether you are preparing for a test or sharpening your fundamental knowledge, these paired definitions will make revision efficient and memorable.
掌握进阶数学的语言是在 OCR Year 10 进阶数学中取得成功的关键。这份速记指南将关键术语拆解为精炼的中英双语解释，帮助你建立信心，避免常见误解。无论你是在准备考试还是巩固基础知识，这些配对定义都能让复习更高效、更难忘。
1. Algebraic Expressions & Polynomials | 代数表达式与多项式
A term is a single mathematical element, such as 7, x, or 4xy. In an expression, terms are separated by + or - signs. A coefficient is the number multiplying the variable; in 3x², 3 is the coefficient. Like terms have the same variable part and can be combined.
项 是单一的数学元素，例如 7、x 或 4xy。在表达式中，项之间由加减号分隔。系数 是乘以变量的数字；在 3x² 中，3 是系数。同类项 具有相同的变量部分，可以合并。
A polynomial is an expression with multiple terms where variables have non‑negative integer exponents, like 2x³ - x + 5. The degree of a polynomial is the highest power of the variable. The leading coefficient is the coefficient of that highest‑degree term. These features determine the graph's end behaviour.
多项式 是含有多个项的表达式，变量的指数为非负整数，例如 2x³ - x + 5。次数 是变量的最高次幂。首项系数 是最高次项的系数。这些特征决定了图像两端的走势。
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Tip: Think of 'coefficient' as the number that 'co‑operates' with the variable, and 'degree' as the height of the power ladder.
提示：将 "coefficient" 想象成与变量 "合作" 的数字，而 "degree" 就是幂次阶梯的高度。
2. Quadratic Equations & the Discriminant | 二次方程与判别式
A quadratic equation has the standard form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola. The discriminant Δ = b² - 4ac 'discriminates' ct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there are no real roots.
二次方程 的标准形式为 ax² + bx + c = 0，且 a ≠ 0。它的图像是一条 抛物线。判别式 Δ = b² - 4ac "判别" 根的性质：若 Δ > 0，有两个不同的实根；若 Δ = 0，有一个重复实根；若 Δ < 0，没有实根。
Δ = b² - 4ac
Completing the square rewrites the quadratic as a(x + p)² + q, which quickly gives the vertex (-p, q) and helps solve equations. The process turns ax² + bx + c into a perfect square plus a constant.
配方法 将二次式改写为 a(x + p)² + q 的形式，能快速给出顶点 (-p, q) 并帮助解方程。该过程将 ax² + bx + c 变成完全平方加常数。
Remember: Discriminant decides the destiny of roots.
记住：判别式决定了根的命运。
3. Binomial Expansion | 二项式展开
Binomial expansion uses the binomial theorem to expand (a + b)ⁿ for positive integer n. The coefficients are given by Pascal's triangle or the combination formula nCr = n! / (r!(n-r)!). Each term in the expansion has the pattern nCr · aⁿ⁻ʳ · bʳ.
二项式展开 利用二项式定理展开 (a + b)ⁿ（n 为正整数）。系数由 帕斯卡三角形 或 组合数公式 nCr = n! / (r!(n-r)!) 给出。展开式中每一项的形式为 nCr · aⁿ⁻ʳ · bʳ。
xy² + ³C₃ y³ = x³ + 3x²y + 3xy² + y³. The powers of a decrease while powers of b increase.
例如，(x + y)³ = ³C₀ x³ + ³C₁ x²y + ³C₂ xy² + ³C₃ y³ = x³ + 3x²y + 3xy² + y³。其中 a 的指数递减，b 的指数递增。
Tip: Pascal's triangle is built by adding the two numbers directly above; nCr is read as 'n choose r'.
提示：帕斯卡三角形的每个数字由上方两数相加而得；nCr 读作 "n 选 r"。
4. Simultaneous Equations | 联立方程
Simultaneous equations are a set of equations that share variables. Solving them means finding values that satisfy all equations at once. For linear equations, use elimination (adding/subtracting to remove a variable) or substitution (replacing one variable with an expression).
联立方程 是一组共享变量的方程。解联立方程意味着找出同时满足所有方程的值。对于线性方程，可用 消元法（通过加减消去一个变量）或 代入法（用一个表达式替换一个变量）。
When one equation is linear and
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