Chapter Review 8: The Binomial Expansion | 第8章复习：二项式展开

📚 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 core skill in Edexcel AS Pure Mathematics. You will revise factorial notation, nCr, Pascal’s triangle, and how to use the expansion formula to find terms and coefficients efficiently. The focus is on building accuracy with signs, powers, and coefficient simplification so that you can handle both standard expansions and exam-style coefficient questions confidently.

本章复习涵盖正整数幂的二项式展开，这是 Edexcel AS 纯数学的核心技能。你将复习阶乘记号、nCr、帕斯卡三角，以及如何高效使用展开公式求项与系数。重点在于训练符号、幂次和系数化简的准确性，帮助你从容应对标准展开和考试中常见的系数问题。

1. Factorials and nCr Notation | 阶乘与 nCr 记号

The factorial of a positive integer n, written n!, is the product of all positive integers from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1, which is essential when simplifying binomial coefficients such as nC0 or nCn.

正整数 n 的阶乘写作 n!，表示从 n 一直乘到 1 的所有正整数之积。例如 5! = 5 × 4 × 3 × 2 × 1 = 120。按照约定，0! = 1，这在化简 nC0 或 nCn 等二项式系数时非常重要。

The binomial coefficient nCr, also read as ‘n choose r’, gives the number of ways to choose r objects from n distinct objects. It is calculated by the formula nCr = n! / [r!(n – r)!]. On most scientific calculators this is the nCr button, but you should also be able to calculate small values by hand.

二项式系数 nCr，读作“n 选 r”，表示从 n 个不同对象中选出 r 个对象的组合数。计算公式为 nCr = n! / [r!(n – r)!]。大多数科学计算器上都有 nCr 按钮，但你仍应会手算较小的值。

For example, 7C3 = 7! / (3!4!) = (7 × 6 × 5) / (3 × 2 × 1) = 35. Notice that the 4! cancels with part of 7!, which makes hand calculation much quicker than expanding the full factorial.

例如，7C3 = 7! / (3!4!) = (7 × 6 × 5) / (3 × 2 × 1) = 35。注意 4! 与 7! 中的部分因子相消，这样手算比完全展开阶乘快得多。

2. Pascal’s Triangle | 帕斯卡三角

Pascal’s triangle is a quick way to read off binomial coefficients for small powers. Each row begins and ends with 1, and every interior entry is the sum of the two entries directly above it. The row for n = 0 is simply 1.

帕斯卡三角是快速读取较小幂次二项式系数的方法。每行首尾都是 1，每个内部数字是正上方两个数字之和。n = 0 的那一行只有 1。

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
n = 5: 1 5 10 10 5 1

The row for n = 5 gives the coefficients for the expansion of (1 + x)⁵: 1, 5, 10, 10, 5, 1. Pascal’s triangle is useful for checking quickly whether your coefficients are sensible, but for larger powers the formula is more efficient.

n = 5 这一行给出 (1 + x)⁵ 展开的系数：1、5、10、10、5、1。帕斯卡三角适合快速检查系数是否合理，但对于较大的幂次，使用公式更高效。

3. The Binomial Expansion Formula | 二项式展开公式

For any positive integer n, the binomial expansion of (a + b)ⁿ is given by the finite sum below. Each term has a binomial coefficient, a descending power of a, and an ascending power of b.

对于任意正整数 n，(a + b)ⁿ 的二项式展开由以下有限和给出。每一项包含一个二项式系数、a 的降幂以及 b 的升幂。

(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + bⁿ

The general term is the (r + 1)th term, written as Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ, where r takes the values 0, 1, 2, …, n. Because r starts at 0, the kth term in the expansion uses r = k – 1. This is a common source of errors if you do not write out the general term first.

通项是第 (r + 1) 项，写作 Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ，其中 r 取 0、1、2、…、n。由于 r 从 0 开始，展开中的第 k 项对应 r = k – 1。如果不先写出通项，这是常见的错误来源。

In the formula, the sum of the exponents in each term is always n: the power of a is n – r and the power of b is r. This symmetry helps you check that you have not missed a term or used the wrong index.

在公式中，每一项的指数之和始终为 n：a 的指数是 n – r，b 的指数是 r。这种对称性可以帮助你检查是否漏项或使用了错误的序号。

4. Expanding (1 + x)ⁿ | 展开 (1 + x)ⁿ

When a = 1 and b = x, the binomial expansion simplifies neatly. The coefficients are exactly the entries from the nth row of Pascal’s triangle, which is why this form is often used to introduce the pattern.

当 a = 1、b = x 时，二项式展开会变得非常简洁。系数恰好是帕斯卡三角第 n 行的数字，这就是为什么常以这种形式引入展开规律。

(1 + x)ⁿ = 1 + n x + [n(n – 1) / 2!] x² + [n(n – 1)(n – 2) / 3!] x³ + … + xⁿ

Example: Expand (1 + x)⁵. Using the fifth row 1, 5, 10, 10, 5, 1 gives:

示例：展开 (1 + x)⁵。使用第五行系数 1、5、10、10、5、1，得到：

(1 + x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵

You should be able to write the first few terms of (1 + x)ⁿ quickly without expanding n! every time. This is especially useful in approximation questions where only the first two or three terms are needed.

你应该能够快速写出 (1 + x)ⁿ 的前几项，而不必每次都展开阶乘。这在只需要前两项或前三项的近似问题中尤其有用。

5. Expanding (a + bx)ⁿ | 展开 (a + bx)ⁿ

Most exam questions use the more general form (a + bx)ⁿ. To expand this, replace b in the standard formula with the whole block bx. Treat bx as a single term, then raise both b and x to the correct power in each step.

大多数考题使用更一般的形式 (a + bx)ⁿ。展开时，将标准公式中的 b 替换为整个块 bx。把 bx 当作一个整体，然后在每一步中让 b 与 x 同时取正确的幂次。

Example: Expand (2 + 3x)³ fully.

示例：完全展开 (2 + 3x)³。

(2 + 3x)³ = 2³ + ³C₁ 2² (3x) + ³C₂ 2¹ (3x)² + (3x)³

= 8 + 3 × 4 × 3x + 3 × 2 × 9x² + 27x³ = 8 + 36x + 54x² + 27x³

A very common mistake is to write only 3x² instead of (3x)² = 9x². Always place brackets around the bx block before applying the power, especially when b is negative or a fraction.

一个非常常见的错误是只写 3x²，而不是 (3x)² = 9x²。在乘方之前，一定要给 bx 块加上括号，尤其是当 b 为负数或分数时更应如此。

6. Finding a Particular Term | 求特定项

Exam questions often ask for a single term, such as ‘find the coefficient of

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Chapter Review 8: The Binomial Expansion | 第8章复习：二项式展开