A-Level Mathematics: Binomial Expansion | A-Level数学：二项式展开

📚 A-Level Mathematics: Binomial Expansion | A-Level数学：二项式展开

The binomial expansion is one of the most powerful tools in A-Level Mathematics. It lets us expand expressions like (a + b)ⁿ without multiplying out brackets term by term, and in Year 2 it extends to fractional and negative powers as infinite series. Mastering this topic is essential for Edexcel Pure Mathematics Papers 1 and 2, and it appears repeatedly in both short and extended problem contexts.

二项式展开是A-Level数学中最强大的工具之一。它让我们无需逐项展开括号就能处理 (a + b)ⁿ 这类表达式，并在第二年延伸到分数次幂和负次幂的无穷级数。掌握这一主题对于Edexcel纯数学试卷1和试卷2至关重要，它在简答题和综合题中反复出现。

1. What Is the Binomial Expansion | 什么是二项式展开

At a basic level, the binomial expansion answers a simple question: how do we write (a + b)ⁿ in expanded form? For small powers we already know the results: (a + b)² = a² + 2ab + b², and (a + b)³ = a³ + 3a²b + 3ab² + b³. But for powers like 7 or 12, manual expansion is impractical.

在最基本的层面上，二项式展开回答了一个简单的问题：我们如何写出 (a + b)ⁿ 的展开形式？对于较小的幂，我们已经知道结果：(a + b)² = a² + 2ab + b²，(a + b)³ = a³ + 3a²b + 3ab² + b³。但对于7次或12次这样的幂，手动展开是不切实际的。

The binomial theorem provides a direct formula. For a positive integer n:

二项式定理提供了一个直接公式。对于正整数 n：

(a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ

Each term has the form C(n,r)aⁿ⁻ʳbʳ, where r runs from 0 to n. The coefficients C(n,r) are called binomial coefficients.

每一项都具有 C(n,r)aⁿ⁻ʳbʳ 的形式，其中 r 从 0 取到 n。系数 C(n,r) 被称为二项式系数。

2. Pascal’s Triangle | 帕斯卡三角

Before the algebraic formula, Pascal’s Triangle offers a visual way to find binomial coefficients. The triangle begins with a single 1 at the top; each row below is formed by adding the two numbers directly above it.

在代数公式之前，帕斯卡三角为我们提供了一种直观寻找二项式系数的方法。这个三角形从顶部的1开始；下面每一行都由其正上方两个数字相加得到。

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

For example, the row 1 3 3 1 corresponds to the coefficients of (a + b)³. Pascal’s Triangle is useful for small values of n, but for larger powers we need a more systematic method using factorials.

例如，1 3 3 1 这一行对应的是 (a + b)³ 的系数。帕斯卡三角对较小的 n 值很有用，但对于较大的幂，我们需要一种使用阶乘的更系统的方法。

3. Factorials and nCr | 阶乘与组合数

The factorial of a positive integer n, written n!, is the product of all positive integers up to n:

正整数 n 的阶乘写作 n!，它是从1到 n 所有正整数的乘积：

n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1

By convention, 0! = 1. The binomial coefficient C(n,r) is defined as:

按照约定，0! = 1。二项式系数 C(n,r) 定义为：

C(n,r) = n! ⁄ (r!(n − r)!)

This is often read as “n choose r” and appears in the Edexcel formula booklet as ⁿCᵣ. Two properties are especially important:

它通常读作“n 选 r”，在Edexcel公式手册中写作 ⁿCᵣ。有两个性质特别重要：

C(n,r) = C(n, n − r), so the coefficients are symmetric.

C(n,r) = C(n, n − r)，即系数是对称的。
C(n,0) = C(n,n) = 1 and C(n,1) = n.

C(n,0) = C(n,n) = 1 且 C(n,1) = n。

For example, C(5,2) = 5! ⁄ (2! × 3!) = 120 ⁄ (2 × 6) = 10. This matches the entry 10 in Pascal’s triangle.

例如，C(5,2) = 5! ⁄ (2! × 3!) = 120 ⁄ (2 × 6) = 10。这与帕斯卡三角中的10一致。

4. The Binomial Theorem | 二项式定理

The binomial theorem combines the pattern of Pascal’s triangle with the powers of a and b:

二项式定理将帕斯卡三角的规律与 a 和 b 的幂结合起来：

(a + b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ, for r = 0 to n

In an exam, it is often easier to use the special form with first term 1. If we write a = 1 and keep x for the second term, we get:

在考试中，使用首项为1的特殊形式往往更方便。如果我们令 a = 1，并用 x 表示第二项，就会得到：

(1 + x)ⁿ = 1 + nx + [n(n − 1)⁄2!]x² + [n(n − 1)(n − 2)⁄3!]x³ + … + xⁿ

This form is especially useful because many questions ask for the expansion of expressions like (1 + 3x)⁶ or (2 − x)⁵, which can be manipulated into this structure.

这种形式特别有用，因为许多题目要求展开类似 (1 + 3x)⁶ 或 (2 − x)⁵ 的表达式，而这些表达式都可以化为此结构。

Take (1 + 3x)⁶. Using the theorem with x replaced by 3x:

以 (1 + 3x)⁶ 为例。用 3x 替换定理中的 x：

(1 + 3x)⁶ = 1 + 6(3x) + 15(3x)² + 20(3x)³ + …

= 1 + 18x + 135x² + 540x³ + …

5. The General Term | 通项公式

To find a specific term without writing out the whole expansion, examiners often expect the general term. In the expansion of (a + b)ⁿ, the term containing bʳ is:

为了不写出整个展开式就能找到特定项，考官通常期望我们使用通项。在 (a + b)ⁿ 的展开式中，含有 bʳ 的项为：

T(r+1) = C(n,r) aⁿ⁻ʳ bʳ

Suppose we want the coefficient of x⁴ in (2 + 3x)⁸. Here a = 2, b = 3x, n = 8. Since we need bʳ = (3x)ʳ to give xʳ, we require r = 4. The term is:

假设我们要找 (2 + 3x)⁸ 中 x⁴ 的系数。这里 a = 2，b = 3x，n = 8。因为需要 bʳ = (3x)ʳ 给出 xʳ，所以要求 r = 4。该项为：

C(8,4)(2)⁴(3x)⁴ = 70 × 16 × 81x⁴ = 90720x⁴

So the coefficient of x⁴ is 90720. Careful handling of powers of 2 and 3 is critical here; many students forget to raise the constant inside the bracket.

因此 x⁴ 的系数是90720。在这里仔细处理2和3的幂至关重要；许多学生忘记对括号内的常数取幂。

6. Negative and Fractional Indices | 负指数与分数指数

In Edexcel Year 1, n must be a positive integer. In Year 2, the theorem is extended so that n can be any rational number. For such values, the expansion becomes an infinite series:

在Edexcel第一年，n 必须是正整数。在第二年，定理被推广到 n 为任意有理数的情况。对于这样的数值，展开式变为无穷级数：

(1 + x)ⁿ = 1 + nx + [n(n − 1)⁄2!]x² + [n(n − 1)(n − 2)⁄3!]x³ + …

For example, expanding (1 + x)⁻¹ gives:

例如，(1 + x)⁻¹ 的展开式为：

(1 + x)⁻¹ = 1 − x + x² − x³ + x⁴ − …

And expanding (1 + x)^{1/2} gives:

而 (1 + x)^{1/2} 的展开式为：

(1 + x)^{1/2} = 1 + (1⁄2)x − (1⁄8)x² + (1⁄16)x³ − …

Notice that the coefficients are now fractions. The pattern of successive coefficients continues forever; there is no final xⁿ term because n is not a positive integer.

注意，系数现在是分数。随后各项系数的规律会无限延续；由于 n 不是正整数，所以不存在最终的 xⁿ 项。

7. The Validity Condition | 收敛条件

Whenever n is negative or fractional, the infinite expansion only converges to the true value of (1 + x)ⁿ when |x| < 1, that is, −1 < x < 1. This is called the interval of validity, and it must be stated in exam answers.

当 n 为负数或分数时，无穷展开式只有在 |x| < 1 即 −1 < x < 1 时才收敛于 (1 + x)ⁿ 的真实值。这称为收敛区间，必须在考试答案中写出。

Why does this condition appear? The binomial series is a type of power series, and power series converge only within their radius of convergence. For (1 + x)ⁿ with rational n, that radius is exactly 1.

为什么会出现这个条件？二项式级数是一种幂级数，而幂级数只在其收敛半径内收敛。对于有理数 n 的 (1 + x)ⁿ，该半径恰好为1。

If the bracket is not of the form (1 + x), we must factor out the constant first. For example, to expand (4 − x)⁻¹, rewrite it as:

如果括号不是 (1 + x) 的形式，我们必须先提取常数因子。例如，要展开 (4 − x)⁻¹，可重写为：

(4 − x)⁻¹ = 4⁻¹(1 − x⁄4)⁻¹

Since the bracket is now (1 + u) with u = −x⁄4, the validity condition is |−x⁄4| < 1, giving |x| < 4.

因为括号现在是 (1 + u)，其中 u = −x⁄4，所以收敛条件是 |−x⁄4| < 1，即 |x| < 4。

8. Using Partial Fractions Before Expansion | 先部分分式再展开

When a rational expression is too complicated to expand directly, we first split it into partial fractions. This is a standard Edexcel Year 2 technique. Consider:

当一个有理表达式过于复杂而无法直接展开时，我们首先将其拆分为部分分式。这是Edexcel第二年的标准技巧。考虑：

(5 − 3x) ⁄ ((1 + x)(2 − x))

We decompose it as:

我们将其分解为：

(5 − 3x)⁄((1 + x)(2 − x)) = A⁄(1 + x) + B⁄(2 − x)

Solving for A and B gives A = 2 and B = 3. Then each fraction is converted into binomial form:

解得 A = 2，B = 3。然后每个分式被转化为二项式形式：

2(1 + x)⁻¹ + 3(2 − x)⁻¹ = 2(1 + x)⁻¹ + (3⁄2)(1 − x⁄2)⁻¹

Each part is then expanded separately, and the two series are added. The validity of the combined result is the intersection of the individual validity intervals: |x| < 1 and |x| < 2, so overall |x| < 1.

然后分别展开每一部分，再将两个级数相加。合并结果的收敛区间是各自收敛区间的交集：|x| < 1 与 |x| < 2，因此总体为 |x| < 1。

Partial fraction approach: always compare the degrees of the numerator and denominator first, and use the correct form of partial fractions for repeated or quadratic factors.

部分分式方法：务必先比较分子和分母的次数，并对重因子或二次因子使用正确的部分分式形式。

9. Numerical Approximations | 数值近似

One practical application of the binomial expansion is estimating roots and powers without a calculator. For example, to approximate √1.04, write:

二项式展开的一个实际应用是在不使用计算器的情况下估算根和幂。例如，要估算 √1.04，可写作：

√1.04 = (1 + 0.04)^{1/2}

Using the fractional-index expansion:

使用分数指数展开式：

(1 + 0.04)^{1/2} ≈ 1 + (1⁄2)(0.04) − (1⁄8)(0.04)²

= 1 + 0.02 − 0.0002 = 1.0198

Since |0.04| < 1, the series converges and the approximation is close to the true value 1.0198039. In exam questions, you may be asked to use three or four terms and then judge the accuracy of the result.

因为 |0.04| < 1，级数收敛，该近似值接近真实值1.0198039。在考试题中，可能会要求你使用三项或四项，然后判断结果的精确度。

10. Common Pitfalls and Exam Tips | 常见错误与考试提示

Students often lose marks in binomial expansion questions for avoidable reasons. The list below covers the most frequent errors in Edexcel exams.

学生常常因为可以避免的原因在二项式展开题目中失分。下面列出的是Edexcel考试中最常见的错误。

Forgetting to state the validity condition |x| < 1 when n is negative or fractional.

当 n 为负数或分数时，忘记写出收敛条件 |x| < 1。
Raising only the variable to a power instead of the whole bracket term, e.g. writing (3x)² as 3x² rather than 9x².

只对变量取幂而不是对整个括号项取幂，例如把 (3x)² 写成 3x² 而不是 9x²。
Using the Year 1 formula C(n,r) when n is not a positive integer; the factorial definition fails, so the infinite series form must be used.

当 n 不是正整数时仍使用第一年的公式 C(n,r)；此时阶乘定义失效，必须使用无穷级数形式。
Sign errors when expanding expressions like (1 − 2x)ⁿ; substitute u = −2x into (1 + u)ⁿ.

展开 (1 − 2x)ⁿ 时出现符号错误；应将 u = −2x 代入 (1 + u)ⁿ。
When using partial fractions, forgetting to find the intersection of validity intervals for the final series.

使用部分分式时，忘记计算最终级数的收敛区间交集。

Always write the general term before jumping to a coefficient, and check that the powers of each constant factor add up correctly. With careful practice, binomial expansion questions become routine and reliable marks.

在直接求系数之前，务必先写出通项，并检查各常数因子的幂是否正确相加。通过仔细练习，二项式展开题会变成常规且稳定得分的题型。

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A-Level Mathematics: Binomial Expansion | A-Level数学：二项式展开