📚 Binomial Expansion | 二项式展开
Binomial expansion is a fundamental technique in A-Level pure mathematics that allows us to write expressions of the form (a + b)n as a sum of terms involving powers of a and b. For positive integer n this gives a finite polynomial; for negative or fractional n it produces an infinite series with important convergence conditions. Edexcel candidates must be confident in expanding both standard forms and in using expansions to find approximations or to simplify algebraic work.
二项式展开是 A-Level 纯数中的一项基本技巧，能够把形如 (a + b)n 的表达式写成一系列含有 a 和 b 幂次的项之和。当 n 为正整数时，得到的是有限多项式；当 n 为负数或分数时，将产生一个具有重要收敛条件的无穷级数。Edexcel 考生必须熟练展开标准形式，并会利用展开式求近似值或简化代数运算。
1. The Binomial Theorem for Positive Integers | 正整数指数的二项式定理
For a positive integer n, the binomial theorem states that (a + b)n = Σr=0n nCr an−r br, where nCr = n! / [r!(n−r)! This formula gives a sum of n+1 terms, widely used to expand powers of binomials quickly.
对于正整数 n，二项式定理指出 (a + b)n = Σr=0n nCr an−r br，其中 nCr = n! / [r!(n−r)!]。该公式给出 n+1 项的和，被广泛用于快速展开二项式的幂。
The coefficients nCr are symmetric: nCr = nCn−r. For example, expanding (x + y)4 gives x4 + 4x3y + 6x2y2 + 4xy3 + y4.
系数 nCr 具有对称性：nCr = nCn−r。例如，展开 (x + y)4 得到 x4 + 4x3y + 6x2y2 + 4xy3 + y4。
2. Pascal’s Triangle | 帕斯卡三角形
Pascal’s triangle is a simple visual tool for obtaining binomial coefficients for small n. Each row corresponds to the coefficients of (a + b)n arting with n = 0 at the top. Every entry is the sum of the two entries directly above it.
帕斯卡三角形是获取小 n 二项式系数的简单直观工具。每一行对应 (a + b)n 的系数，顶端从 n = 0 开始。每个数字是它正上方两个数字之和。
The first five rows are shown below:
| n=0 | 1 | ||||
| n=1 | 1 | 1 | |||
| n=2 | 1 | 2 | 1 | ||
| n=3 | 1 | 3 | 3 | 1 | |
| n=4 | 1 | 4 | 6 | 4 | 1 |
Although Pascal’s triangle is helpful for small expansions, the factorial formula becomes essential for larger n or when the power is not a positive integer.
虽然帕斯卡三角形对小型展开很有帮助，但当 n 较大或指数不是正整数时，阶乘公式就变得必不可少。
3. Factorial and Combination Notation | 阶乘与组合符号
The binomial coefficient nCr is read as 'n choose r' and can be computed using factorials: nCr = n! / (r! (n−r)!). For example, 5C2 = 5!/(2!3! ncy in calculating these coefficients both manually and using a calculator’s nCr function.
二项式系数 nCr 读作“n 选 r”，可以用阶乘计算：nCr = n! / (r! (n−r)!)。例如 5C2 = 5!/(2!3!) = 10。Edexcel 考试要求学生能熟练手工计算这些系数，也会使用计算器上的 nCr 功能。
It is crucial to remember that 0! = 1 and nC0 = nCn = 1. The symmetry property often simplifies expansion work.
必须牢记 0! = 1，且 nC0 = nCn = 1。对称性质常常能简化展开工作。
4. Expanding (a + b)n in Standard Form | 展开标准形式 (a + b)n
To expand (a + b)n, write the general term Tr+1 = nCr an−r br n. A common exam question asks for the term independent of x, or for a specific coefficient.
展开 (a + b)n 时，写出通项 Tr+1 = nCr an−r br。从 r = 0 开始，一直增加到 r = n。常见考题会要求找出与 x 无关的项或者特定项的系数。
For instance, in (2x + 3)5, the term in x3 arises when r = 2: 5C2 (2x)3 (3)2 = 10 × 8x3 × 9 = 720x3. Straightforward application of the formula avoids algebraic slips.
例如，在 (2x + 3)5 中，含 x3 的项出现在 r = 2 时：5C2 (2x)3 (3)2 = 10 × 8x3 × 9 = 720x3。直接套用公式可避免代数失误。
5. The General Binomial Expansion (Negative and Fractional Powers) | 一般二项式展开（负指数与分数指数）
When n is not a positive integer, the expansion of (1 + x)n becomes an infinite series: (1 + x)n = 1 + nx + n(n−1)/2! x2 + n(n−1)(n−2)/3! x3 + ... , valid for |x| < 1. This extension is central to the Edexcel A-Level syllabus.
当 n 不是正整数时，(1 + x)n 的展开式变成一个无穷级数：(1 + x)n = 1 + nx + n(n−1)/2! x2 + n(n−1)(n−2)/3! x3 + ... ，适用于 |x| < 1。这一推广是 Edexcel A-Level 大纲的核心内容。
The coefficients no longer come from the nCr formula for integer r, because n is not an integer. Instead, the series uses a generalised binomial coefficient: nCr = n(n−1)...(n−r+1) / r!. You must be careful with signs when n is negative.
此时系数不再来自整数 r 的 nCr 公式，因为 n 不是整数。级数使用推广的二项式系数：nCr = n(n−1)...(n−r+1) / r!。当 n 为负数时，必须注意符号。
Example: Expand (1 + 2x)−1 up to x3. Here n = −1, so (1 + 2x)−1 = 1 + (−1)(2x) + (−1)(−2)/2! (2x)2 + (−1)(−2)(−3)/3! (2x)3 + ... = 1 − 2x + 4x2 − 8x3 + ... , |2x| < 1 → |x| < ½.
例子：展开 (1 + 2x)−1 至 x3 项。这里 n = −1，因此 (1 + 2x)−1 = 1 + (−1)(2x) + (−1)(−2)/2! (2x)2 + (−1)(−2)(−3)/3! (2x)3 + ... = 1 − 2x + 4x2 − 8x3 + ... ，有效范围 |2x| < 1 即 |x| < ½。
6. Handling (a + bx)n When a ≠ 1 | 处理 a ≠ 1 时的 (a + bx)n
For expressions like (4 + 3x)½, you must first factor out the constant to obtain the standard (1 + something) form. Write (4 + 3x)½ = 4½ (1 + ¾x)½ = 2 (1 + ¾x)½, then expand the binomial part.
对于 (4 + 3x)½ 这类表达式，你必须先提取常数因子，得到标准的 (1 + 某量) 形式。写作 (4 + 3x)½ = 4½ (1 + ¾x)½ = 2 (1 + ¾x)½，然后展开二项式部分。
This step is frequently tested, as failing to factor leads to an invalid expansion. The series expansion must always be applied to something of the form (1 + u)n with |u| < 1.
这一步常被考查，因为如果不提取因子会导致无效展开。级数展开始终只能应用于形如 (1 + u)n 且满足 |u| < 1 的形式。
7. Validity Condition and Convergence | 有效条件与收敛性
The infinite binomial expansion for (1 + x)n converges only when |x| < 1. If the question gives a composite expression like (1 + 5x)−2, the validity requirement becomes |5x| < 1, i.e. |x| < 0.2. Stating the range of validity is often worth a mark in Edexcel papers.
(1 + x)n 的无穷二项展开式仅在 |x| < 1 时收敛。如果题目给出复合表达式如 (1 + 5x)−2，有效条件就变成 |5x| < 1，即 |x| < 0.2。给出有效性范围在 Edexcel 试卷中通常占有一分。
For expansions in powers of x that are ascending, the series is an approximation tool; the modulus condition ensures the terms get smaller and the truncation error can be controlled.
对于按 x 升幂排列的展开式，这个级数是一个近似工具；模条件保证各项逐渐变小，截断误差得以控制。
8. Using the Expansion to Find Approximations | 利用展开式求近似值
A common application is to approximate values of roots or reciprocals. For instance, expand (1 + x)½ to x3 and substitute a small x to estimate √1.02. With x = 0.02, √1.02 ≈ 1 + ½(0.02) − ⅛(0.02)2 + 1/16 (0.02)3.
一个常见应用是近似求根或倒数的值。例如，展开 (1 + x)½ 至 x3 项，代入一个小的 x 来估计 √1.02。取 x = 0.02，√1.02 ≈ 1 + ½(0.02) − ⅛(0.02)2 + 1/16 (0.02)3。
You may also be asked to find the percentage error when using only the first two or three terms. Such questions test both expansion skills and understanding of approximation quality.
你也可能被要求计算仅用前两项或前三项时的百分比误差。这类题目既考查展开技能，也考查对近似质量的理解。
9. Partial Fractions and Binomial Expansion | 部分分式与二项式展开
Edexcel frequently combines algebraic partial fractions with binomial expansion. A rational function like (3x+5)/[(1−x)(1+2x)] can be split into partial fractions: A/(1−x) + B/(1+2x), and each term expanded binomially as A(1−x)−1 + B(1+2x)−1. The expansions are valid when |x| < 1 and |2x| < 1 respectively, so the overall validity is |x| < ½.
Edexcel 常将代数部分分式与二项式展开结合考查。形如 (3x+5)/[(1−x)(1+2x)] 的有理函数可拆成部分分式：A/(1−x) + B/(1+2x)，然后分别用二项式展开为 A(1−x)−1 + B(1+2x)−1。展开式分别在 |x| < 1 和 |2x| < 1 时有效，因此整体有效范围为 |x| < ½。
This technique greatly extends the usefulness of binomial expansion and is a regular feature of the Pure Mathematics 2/3 papers.
这一技巧极大地扩展了二项式展开的应用范围，是纯数 2/3 试卷的常见内容。
10. Dealing with Two Variables and Cross Terms | 处理双变量与交叉项
When expanding products of binomials, such as (1 + 2x)3 (1 − x)−1, you can expand each factor separately to the required power and then multiply the series, collecting like terms up to the desired degree. Take care with the interaction between terms.
当展开二项式乘积时，如 (1 + 2x)3 (1 − x)−1，可以分别将每个因式展开到所需幂次，再将级数相乘，合并到所需次数。务必注意项与项之间的相互作用。
Exam questions might ask 'expand up to and including the term in x2', so you must discard contributions beyond that order while multiplying.
考题可能要求“展开到并包括 x2 项”，因此相乘时必须舍去超出该阶的贡献。
11. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
Forgetting to apply the modulus condition for validity: always identify the restriction on x derived from the binomial part.
忘记应用有效性的模条件：始终要确定从二项式部分导出的 x 的限制。
Sign errors when expanding negative powers: write the coefficient as n(n−1)(n−2)... / r! carefully, treating n as negative.
展开负数幂时的符号错误：仔细将系数写作 n(n−1)(n−2)... / r!，将 n 当作负数处理。
Omitting the factorial denominators: the series is NOT 1 + nx + n(n−1)x2 + ...; division by r! is essential.
遗漏阶乘分母：级数不是 1 + nx + n(n−1)x2 + ...；除以 r! 是必不可少的。
Applying the finite nCr formula to fractional n: use the product form instead.
将有限 nCr 公式用于分数 n：应改用连乘形式。
12. Summary and Exam Tips | 总结与考试技巧
Binomial expansion is a versatile tool throughout the Edexcel A-Level course. Master both the finite expansion for positive integer n and the infinite series form for rational n. Always rewrite the binomial into (1 + u)n before expanding, and state the range of x for which the expansion is valid. Practice reading questions carefully: they may ask for a specific term, an approximation, or even a proof of identity.
二项式展开是贯穿 Edexcel A-Level 课程的多功能工具。既要掌握正整数 n 的有限展开，也要掌握有理数 n 的无穷级数形式。展开前一定要将二项式改写为 (1 + u)n 的形式，并说明展开有效的 x 范围。练习仔细审题：题目可能要求求某一特定项、一个近似值甚至是恒等式证明。
In the exam, show clear working for each coefficient, and remember that the validity mark is an easy mark to pick up. With regular practice, binomial expansion becomes one of the most straightforward and rewarding topics on the paper.
考试中，每个系数的计算要展示清晰步骤，并记住有效性这一分很容易获得。通过经常练习，二项式展开将成为试卷上最直接且最有把握得分的题目之一。
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