📚 Chapter Review 8: The Binomial Expansion | 第8章复习：二项式展开
This chapter review covers the binomial expansion for positive integer powers, a key topic in Edexcel A-Level Pure Mathematics. You will learn how to expand expressions of the form (a + b)n using the binomial theorem, apply factorial notation and nCr, handle specific terms, and use expansions for estimation. Each section is designed to strengthen your skills for the AS and A-Level exams, with bilingual explanations that mirror the textbook approach.
本章复习涵盖正整数幂的二项式展开，这是 Edexcel A-Level 纯数学的重要考点。你将学习如何使用二项式定理展开 (a + b)n 形式的表达式，运用阶乘记法与组合数 nCr，求解特定项，并利用展开式进行估值。每一节都旨在强化你的 AS 与 A-Level 应考能力，中英对照讲解紧扣教材思路。
1. What is the Binomial Expansion? | 什么是二项式展开？
A binomial is an algebraic expression containing two terms, such as (x + y) or (2a - 3b). The binomial expansion is a way to write (a + b)n as a sum of terms involving powers of a and b, where n is a positive integer. For small values of n, you can multiply out the brackets manually, but the binomial theorem gives a systematic method for any positive integer n.
二项式是指含有两项的代数式，例如 (x + y) 或 (2a - 3b)。二项式展开就是将 (a + b)n 写成关于 a 和 b 的幂次之和的方法，其中 n 为正整数。当 n 较小时，可以手工逐项相乘，但二项式定理提供了一种适用于任意正整数 n 的系统性方法。
2. Pascal’s Triangle and Simple Expansions | 帕斯卡三角与简单展开
Pascal's triangle is a triangular array where each number is the sum of the two directly above it. The row corresponding to power n gives the coefficients of the expansion of (1 + x)n. For example, the 3rd row (starting from row 0) is 1, 3, 3, 1, which matches (1 + x)3 = 1 + 3x + 3x2 + x3. This method is quick for small powers but becomes impractical for large n.
帕斯卡三角是一个三角形排列，其中每个数等于它上方两数之和。与幂次 n 对应的行给出了 (1 + x)n 展开式的系数。例如第 3 行（从第 0 行开始）是 1, 3, 3, 1，正好匹配 (1 + x)3 = 1 + 3x + 3x2 + x3。这种方法对低次幂很便捷，但当 n 较大时就不实用了。
3. Factorial Notation and nCr | 阶乘记法与组合数
binomial coefficient nCr (also written as C(n, r) or binomial n choose r). The factorial n! is the product of all positive integers up to n: n! = n × (n-1) × … × 1. The binomial coefficient is defined as nCr = n! / [r! (n - r)!], which gives the coefficient of the term containing br in the expansion of (a + b)n.
为了处理较大的指数，我们使用阶乘记法和二项式系数 nCr（也记作 C(n, r) 或二项式选择）。阶乘 n! 是所有正整数到 n 的乘积：n! = n × (n-1) × … × 1。二项式系数定义为 nCr = n! / [r! (n - r)!]，它给出了 (a + b)n 展开式中含有 br 那一项的系数。
4. The Binomial Theorem Formula | 二项式定理公式
For a positive integer n, the binomial theorem states:
(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, r = 0, 1, …, n
This means the expansion is nC0 aⁿ b⁰ + nC1 aⁿ⁻¹ b¹ + nC2 aⁿ⁻² b² + … + nCn a⁰ bⁿ. Each term combines a decreasing power of a and an increasing power of b, with coefficient nCr. The total number of terms is n + 1.
对于正整数 n，二项式定理陈述如下：
(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, r = 0, 1, …, n
这意味着展开式为 nC0 aⁿ b⁰ + nC1 aⁿ⁻¹ b¹ + nC2 aⁿ⁻² b² + … + nCn a⁰ bⁿ。每一项中 a 的幂次递减，b 的幂次递增，系数为 nCr。总项数为 n + 1。
5. Expanding (a + b)n | 展开 (a + b)n
When expanding expressions like (2x + 3)4 h nCr values: 4C0 = 1, 4C1 = 4, 4C2 = 6, 4C3 = 4, 4C4 = 1. Then substitute carefully: (2x)4 + 4(2x)3(3) + 6(2x)2(3)2 + 4(2x)(3)3 + 34. Simplify each term to get 16x4 + 96x3 + 216x2 + 216x + 81. Always check that powers of a decrease and powers of b increase.
展开 (2x + 3)4 这类式子时，首先确认 a = 2x，b = 3，n = 4。使用公式并代入 nCr 值：4C0 = 1，4C1 = 4，4C2 = 6，4C3 = 4，4C4 = 1。然后仔细代入：(2x)4 + 4(2x)3(3) + 6(2x)2(3)2 + 4(2x)(3)3 + 34。逐项化简得到 16x4 + 96x3 + 216x2 + 216x + 81。务必检查 a 的幂次递减、b 的幂次递增。
6. Expanding (1 + x)n | 展开 (1 + x)n 及应用
A special case is (1 + x)n and b = x. Since 1 raised to any power is 1, the expansion simplifies to 1 + nC1 x + nC2 x2 + … + xn. This form is particularly useful for approximations and when the coefficient of a is 1. For example, (1 + 2y)5 = 1 + 5(2y) + 10(4y2) + 10(8y3) + 5(16y4) + 32y5, which simplifies to 1 + 10y + 40y2 + 80y3 + 80y4 + 32y5.
一个特例是 (1 + x)n，此时 a = 1，b = x。因为 1 的任何次方仍是 1，展开式简化为 1 + nC1 x + nC2 x2 + … + xn。这种形式在估值以及 a 的系数为 1 时特别有用。例如，(1 + 2y)5 = 1 + 5(2y) + 10(4y2) + 10(8y3) + 5(16y4) + 32y5，化简得 1 + 10y + 40y2 + 80y3 + 80y4 + 32y5。
7. Finding a Specific Term | 特定项求解
You may be asked to find a particular term, such as the term in x3, without writing the full expansion. The general term (the (r+1)th term) is given by nCr an-r br. To find the term in xk, set the power of x equal to k, solve for r, then compute the coefficient. For instance, in (2x - 1/x)6, the term independent of x (constant term) requires the power of x to be zero. The general term is 6Cr (2x)6-r (-1/x)r = 6Cr 26-r (-1)r x6-2r. Setting 6-2r = 0 gives r = 3, so the constant term is 6C3 × 23 × (-1)3 = 20 × 8 × (-1) = -160.
考试中常要求不用完整展开而直接求出某一特定项，例如含 x3 的项。通项（第 r+1 项）公式为 nCr an-r br。要求 xk 的项时，令 x 的指数等于 k，解出 r，再计算系数。例如在 (2x - 1/x)6 中，常数项（与 x 无关）要求 x 的指数为零。通项为 6Cr (2x)6-r (-1/x)r = 6Cr 26-r (-1)r x6-2r。令 6-2r = 0 得 r = 3，因此常数项为 6C3 × 23 × (-1)3 = 20 × 8 × (-1) = -160。
8. Using Binomial Expansion for Approximations | 利用二项展开式进行估算
When n is a small positive integer and the second term is much smaller than 1, you can use the first few terms of a binomial expansion to estimate a value. For example, (1.002)5 can be written as (1 + 0.002)5. Expanding gives 1 + 5(0.002) + 10(0.002)2 + … vides a quick approximation without a calculator. Remember to only take enough terms to reach the required accuracy.
当 n 为较小的正整数且第二项远小于 1 时，可以利用二项展开式的前几项来估算数值。例如，(1.002)5 可以写成 (1 + 0.002)5。展开得 1 + 5(0.002) + 10(0.002)2 + … ≈ 1 + 0.01 + 0.00004 = 1.01004。这提供了一种无需计算器的快速估值方法。记住只取足够多的项以满足精度要求。
9. Binomial Expansion with Terms in Descending / Ascending Powers | 按降幂或升幂展开
Exam questions sometimes ask for the expansion in ascending powers of x, meaning from the lowest to the highest exponent. The binomial theorem naturally gives terms in descending powers of a and ascending powers of b, so when a contains x and b is a constant, the expansion is already in descending powers of x. To write it in ascending powers, reverse the order. Alternatively, if b contains x, the standard expansion may already be in ascending powers of x. Always read the question carefully to present your answer correctly.
考题有时要求按 x 的升幂展开，即从最低次幂排到最高次幂。二项式定理自然给出 a 的降幂、b 的升幂，因此当 a 含 x 而 b 为常数时，展开式已经是 x 的降幂排列。要写成升幂，只需颠倒顺序。反之，若 b 含 x，则标准展开可能已经按 x 升幂排列。务必仔细审题，按要求正确呈现答案。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students forget to apply the power to both the coefficient and the variable inside a term, for example writing (2x)3 as 2x3 instead of 8x3. Another common error is miscalculating nCr: use the nCr button on your calculator or the formula carefully. When dealing with negative signs, ensure you track the sign with each power, e.g., (-y)2 = y2 but (-y)3 = -y3. For approximations, do not truncate too early if high precision is required. In the exam, show the substitution of nCr clearly and write the expression in a structured way before simplifying.
许多学生忘记将幂次施加给括号内的系数和变量，例如把 (2x)3 写成 2x3 而不是 8x3。另一个常见错误是算错 nCr：务必使用计算器的 nCr 键或小心代入公式。处理负号时，要逐项追踪符号，例如 (-y)2 = y2 但 (-y)3 = -y3。进行估值时，如果需要高精度，不要过早截断。考试中，应清晰写出 nCr 的代入过程，并在化简前按结构化格式列出展开式。
11. Worked Example: Expanding (3 - 2x)5 | 例题精讲：展开 (3 - 2x)5
Let's work through a typical exam problem. Expand (3 - 2x)5 in ascending powers of x up to the term in x3. Using n = 5, a = 3, b = -2x. The first few terms are 5C0 (3)5 + 5C1 (3)4 (-2x) + 5C2 (3)3 (-2x)2 + 5C3 (3)2 (-2x)3 efficients: 5C0 = 1, 5C1 = 5, 5C2 = 10, 5C3 = 10. So we get 1×243 + 5×81×(-2x) + 10×27×(4x2) + 10×9×(-8x3) = 243 - 810x + 1080x2 - 720x3. The ascending order is already satisfied because the constant term is first. This example illustrates handling negative b and carefully simplifying powers.
我们来完整解一道典型考题。按 x 的升幂展开 (3 - 2x)5 至 x3 项。这里 n = 5，a = 3，b = -2x。前几项为 5C0 (3)5 + 5C1 (3)4 (-2x) + 5C2 (3)3 (-2x)2 + 5C3 (3)2 (-2x)3。计算系数：5C0 = 1，5C1 = 5，5C2 = 10，5C3 = 10。于是得到 1×243 + 5×81×(-2x) + 10×27×(4x2) + 10×9×(-8x3) = 243 - 810x + 1080x2 - 720x3。升幂排列自然满足，因为常数项在最前。这个例子示范了如何处理负的 b 并仔细化简各次幂。
12. Connection to Algebraic Division and Proof | 与代数除法及证明的联系
Binomial expansions can appear in algebraic fraction simplification or proof questions. For example, expanding (1 + x)n and comparing coefficients helps prove identities involving combinations. Moreover, in the pure maths syllabus, the expansion for positive integer n lays the groundwork for the infinite binomial series in Year 2, where n can be a fraction or negative number. Understanding the positive integer case thoroughly makes the extension to rational powers much smoother.
二项式展开还会出现在代数分式化简或证明题中。例如，展开 (1 + x)n 并比较系数有助于证明涉及组合数的恒等式。此外，在纯数课程中，正整数 n 的展开式为第二学年中分数或负数指数的无穷二项级数打下基础。透彻理解正整数情形，可以使有理数指数的推广变得顺利许多。
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