The Binomial Expansion | 二项式展开

📚 The Binomial Expansion | 二项式展开

The binomial expansion is a core A-Level Pure Mathematics topic in Edexcel. It allows you to expand expressions of the form (a + b)ⁿ, first for positive integer powers using factorial notation and nCr, and later for rational or negative powers using an infinite binomial series. This article covers the key definitions, standard forms, validity conditions, approximation techniques, and common exam traps.

二项式展开是 Edexcel 爱德思 A-Level 纯数学的核心内容。它帮助你将 (a + b)ⁿ 形式的式子展开，先用于正整数指数，使用阶乘记号与 nCr；之后推广到有理数或负数指数，使用无穷二项级数。本文涵盖关键定义、标准形式、有效范围、近似方法以及常见考试陷阱。

1. Pascal’s Triangle and Factorial Notation | 帕斯卡三角形与阶乘记号

For a positive integer n, the coefficients in the expansion of (a + b)ⁿ can be read from Pascal’s triangle. Each row corresponds to one value of n, and each entry is obtained by adding the two adjacent entries directly above it. Factorial notation is defined by n! = n × (n − 1) × … × 2 × 1, with the special case 0! = 1.

对于正整数 n，(a + b)ⁿ 展开中的系数可以从帕斯卡三角形中读出。每一行对应一个 n 的值，每个数由上一行相邻两个数相加得到。阶乘记号定义为 n! = n × (n − 1) × … × 2 × 1，并特别规定 0! = 1。

n	Pascal row / 帕斯卡行
0	1
1	1 1
2	1 2 1
3	1 3 3 1
4	1 4 6 4 1
5	1 5 10 10 5 1

The triangle is useful for small values of n, but for larger powers the nCr formula is much faster and more reliable.

对于较小的 n，帕斯卡三角形很方便；但对于较大的指数，使用 nCr 公式会更快、更可靠。

2. The nCr Formula and Symmetry | nCr 公式与对称性

The binomial coefficient nCr, also written as C(n, r) or n choose r, gives the coefficient of the r-th term. It is calculated by the formula:

二项式系数 nCr，也写作 C(n, r) 或 n choose r，表示第 r 项的系数。其计算公式为：

nCr = n! / (r!(n − r)!)

This coefficient counts the number of ways to choose r objects from n objects without regard to order. It also has a symmetry property:

该系数表示从 n 个对象中不考虑顺序地选出 r 个对象的方法数。它还具有对称性：

nCr = nC(n − r)

For example, 5C2 = 5! / (2!3!) = 10, and 5C3 is also 10. This symmetry can save time when extracting coefficients in an expansion.

例如，5C2 = 5! / (2!3!) = 10，而 5C3 也等于 10。这种对称性在展开式中提取系数时可以节省时间。

3. Expanding (a + b)ⁿ for Positive Integers | 正整数指数下 (a + b)ⁿ 的展开

When n is a positive integer, the binomial expansion is finite and has n + 1 terms. The general form is:

当 n 为正整数时，二项展开是有限的，共有 n + 1 项。其一般形式为：

(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + ⁿCₙ bⁿ

Here the powers of a decrease from n to 0, while the powers of b increase from 0 to n. The sum of the two powers in every term is always n.

这里 a 的指数从 n 逐渐降到 0，而 b 的指数从 0 逐渐升到 n。每一项中两个指数的和始终等于 n。

Example: expand (2 + 3x)⁴.

示例：展开 (2 + 3x)⁴。

(2 + 3x)⁴ = ⁴C₀ 2⁴ + ⁴C₁ 2³(3x) + ⁴C₂ 2²(3x)² + ⁴C₃ 2(3x)³ + ⁴C₄ (3x)⁴

= 16 + 96x + 216x² + 216x³ + 81x⁴

Always check that the number of terms is n + 1, and that the coefficients match the correct row of Pascal’s triangle.

务必检查项数是否为 n + 1，并确认系数与帕斯卡三角形中对应行一致。

4. General Term and Coefficient Extraction | 通项与系数提取

Rather than expanding every term, you can write down the general term directly. For the expansion of (a + b)ⁿ, the (r + 1)-th term is:

你不必展开所有项，可以直接写出通项。对于 (a + b)ⁿ 的展开，第 (r + 1) 项为：

Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ

This is especially useful when a question asks for a particular coefficient, such as the term independent of x or the coefficient of x².

这在题目要求求某一特定系数时尤其有用，例如与 x 无关的项或 x² 的系数。

Example: find the term independent of x in (2x + 1/x)⁶.

示例：求 (2x + 1/x)⁶ 中与 x 无关的项。

Tᵣ₊₁ = ⁶Cᵣ (2x)⁶⁻ʳ (1/x)ʳ = ⁶Cᵣ 2⁶⁻ʳ x⁶⁻²ʳ

For the term independent of x, set 6 − 2r = 0, so r = 3. The term is:

要求与 x 无关的项，令 6 − 2r = 0，因此 r = 3。该项为：

⁶C₃ 2³ = 20 × 8 = 160

This method avoids writing out all seven terms and is essential for efficient exam work.

这种方法避免了写出全部七项，对于高效解题至关重要。

5. Binomial Expansion for Rational Powers | 有理数幂的二项式展开

When n is not a positive integer, the binomial expansion becomes an infinite series. For |x| < 1, the standard series is:

当 n 不是正整数时，二项展开变为无穷级数。当 |x| < 1 时，标准级数为：

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

If n is a positive integer, this series stops after the xⁿ term. If n is rational, negative, or a fraction, the series continues forever, so you usually only need the first few terms.

如果 n 是正整数，该级数在 xⁿ 项之后停止。如果 n 是有理数、负数或分数，级数将无限继续，因此通常只需要前几项。

Example: expand (1 + x)⁻¹ up to x³.

示例：将 (1 + x)⁻¹ 展开到 x³。

(1 + x)⁻¹ = 1 − x + x² − x³ + …

Example: expand (1 + 2x)^(1/2) up to x³.

示例：将 (1 + 2x)^(1/2) 展开到 x³。

(1 + 2x)^(1/2) = 1 + x − (1/2)x² + (1/2)x³ + …

Notice that the coefficients are no longer integers, and the expansion must include a validity statement.

注意系数不再是整数，而且展开式必须附上有效范围说明。

6. Validity Conditions | 有效范围与收敛条件

The infinite binomial series for (1 + x)ⁿ is only valid when |x| < 1. This is called the radius of convergence. If the expression is not in the standard form, you must first factor out the constant.

(1 + x)ⁿ 的无穷二项级数仅在 |x| < 1 时有效。这被称为收敛半径。如果表达式不是标准形式，你必须先提取常数。

For (a + bx)ⁿ, rewrite it as:

对于 (a + bx)ⁿ，先改写为：

(a + bx)ⁿ = aⁿ (1 + (b/a)x)ⁿ

The expansion is then valid for |(b/a)x| < 1, which gives:

展开式在 |(b/a)x| < 1 时有效，即：

|x| < |a/b|

Example: expand (4 + 3x)^(1/2).

示例：展开 (4 + 3x)^(1/2)。

(4 + 3x)^(1/2) = 4^(1/2) (1 + 3x/4)^(1/2) = 2 (1 + 3x/4)^(1/2)

The expansion is valid for |3x/4| < 1, so |x| < 4/3.

展开式在 |3x/4| < 1 时有效，因此 |x| < 4/3。

Students often lose marks by forgetting to state the validity condition or by using the wrong inequality sign.

学生常常因为忘记说明有效范围或使用错误的不等号而丢分。

7. Expanding (a + bx)ⁿ Using Standard Form | 将 (a + bx)ⁿ 化为标准形式展开

To expand (a + bx)ⁿ for rational n, always make the first term inside the bracket equal to 1. Factor out aⁿ first, then apply the standard series to (1 + (b/a)x)ⁿ.

要对有理数 n 展开 (a + bx)ⁿ，总是要使括号内的首项为 1。先提取 aⁿ，然后对 (1 + (b/a)x)ⁿ 使用标准级数。

Example: expand (4 + 3x)^(1/2) up to x².

示例：将 (4 + 3x)^(1/2) 展开到 x²。

(4 + 3x)^(1/2) = 2 (1 + 3x/4)^(1/2)

= 2 [1 + (1/2)(3x/4) + ((1/2)(−1/2)/2!)(3x/4)² + …]

= 2 + (3/4)x − (9/64)x² + …

The same method works for negative powers, such as (2 − x)⁻². You must be careful with signs when substituting the negative power.

同样的方法也适用于负数指数，例如 (2 − x)⁻²。在代入负指数时，你必须特别注意符号。

8. Approximations Using Binomial Series | 利用二项式级数做近似

One of the main applications of the binomial expansion is approximate calculation. If a quantity can be written as (1 + small number)ⁿ, the first few terms give a very good estimate.

二项式展开的主要应用之一是近似计算。如果一个量可以写成 (1 + 小数)ⁿ 的形式，前几项就能给出很好的估计。

Example: approximate √(1.02).

示例：近似计算 √(1.02)。

√(1.02) = (1 + 0.02)^(1/2)

≈ 1 + (1/2)(0.02) − (1/8)(0.02)²

= 1 + 0.01 − 0.00005 = 1.00995

The exact value is about 1.00995049, so the approximation is extremely accurate. In exam questions, you may be asked to state whether the expansion gives an overestimate or an underestimate.

精确值约为 1.00995049，因此这个近似非常准确。在考试题中，可能会问你该展开式给出的是高估值还是低估值。

9. Harder Problems: Two Expansions and Partial Fractions | 较难问题：两次展开与部分分式

Some Edexcel questions require you to split an algebraic fraction into partial fractions first, and then expand each fraction as a binomial series. This is a common Pure 3 or Pure 4 skill.

一些爱德思题目要求你先把代数分式拆成部分分式，然后再把每个分式分别展成二项级数。这是 Pure 3 或 Pure 4 中的常见技能。

Example: expand 3 / [(1 + x)(2 + x)] up to x².

示例：将 3 / [(1 + x)(2 + x)] 展开到 x²。

更多咨询请联系16621398022（同微信）

The Binomial Expansion | 二项式展开