📚 The Binomial Expansion | 二项式展开
The binomial expansion is one of the most important results in A-Level mathematics. It allows us to expand expressions of the form (a+b)ⁿ without laborious multiplication. More remarkably, with a generalised version, we can even expand (1+x)ⁿ for any rational index n, turning algebraic expressions into infinite series that approximate functions.
二项式展开是 A-Level 数学中最重要的结果之一。它使我们无需冗长的乘法即可展开形如 (a+b)ⁿ 的表达式。更奇妙的是，通过推广，我们还可以将任何有理数指数 n 的 (1+x)ⁿ 展开为无穷级数，从而逼近函数值。
1. The Binomial Theorem for Positive Integer n | 正整数 n 的二项式定理
For a positive integer n, the binomial theorem states that:
对正整数 n，二项式定理表述如下：
(a+b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ
where ⁿCₖ = n! / (k!(n−k)! is called the binomial coefficient. This gives a finite expansion with exactly n+1 terms.
其中 ⁿCₖ = n! / (k!(n−k)!) 称为二项式系数。该展开是有限的，共有 n+1 项。
2. Factorial Notation | 阶乘记号
Factorials are defined as n! = n × (n−1) × … × 2 × 1, with 0! simplify many algebraic expressions and are essential for computing binomial coefficients.
阶乘定义为 n! = n × (n−1) × … × 2 × 1，并规定 0! = 1。阶乘简化了许多代数表达式，也是计算二项式系数的基础。
ⁿCₖ = ⁿ! / (k!(n−k)!), ⁴C₂ = 4! / (2! 2!) = 6
Notice that ⁿCₖ = ⁿCₙ₋ₖ, a symmetry that often saves time.
注意 ⁿCₖ = ⁿCₙ₋ₖ，这个对称性常可节省计算时间。
3. Pascal's Triangle | 杨辉三角
Pascal's triangle provides a quick way to generate binomial coefficients for small n. Each row begins and ends with 1, and each interior number is the sum of the two numbers above it.
杨辉三角为较小的 n 提供了一种快速生成二项式系数的方法。每一行以 1 开始和结束，而每个内部数字等于其上方两个数字之和。
| 1 | ||||
| 1 | 1 | |||
| 1 | 2 | 1 | ||
| 1 | 3 | 3 | 1 | |
| 1 | 4 | 6 | 4 | 1 |
Using Pascal's triangle, we can immediately write (1+x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.
利用杨辉三角，我们可以直接写出 (1+x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴。
4. The General Term | 通项
bᵏ. This formula is useful when we need a particular term without writing out the whole expansion.
在 (a+b)ⁿ 的展开中，含 bᵏ 的项为 Tₖ₊₁ = ⁿCₖ aⁿ⁻ᵏ bᵏ。当我们需要特定某项而无需写出整个展开时，这个公式很有用。
Tₖ₊₁ = ⁿCₖ aⁿ⁻ᵏ bᵏ
le, to find the x³ term in (2+3x)⁵, set k=3: T₄ = ⁵C₃ (2)² (3x)³ = 10 × 4 × 27x³ = 1080x³.
例如，要求 (2+3x)⁵ 中 x³ 项的系数，令 k=3：T₄ = ⁵C₃ (2)² (3x)³ = 10 × 4 × 27x³ = 1080x³。
5. Binomial Coefficients and Their Properties | 二项式系数及其性质
The coefficients ⁿCₖ are integers with several important properties:
系数 ⁿCₖ 是整数，具有几个重要性质：
- Symmetry: ⁿCₖ = ⁿCₙ₋ₖ
- Addition rule: ⁿCₖ + ⁿCₖ₊₁ = ⁿ⁺¹Cₖ₊₁
- Sum of coefficients: Σₖ₌₀ⁿ ⁿCₖ = 2ⁿ
These properties are used in proofs, probability, and algebra.
这些性质在证明、概率和代数中都有应用。
6. The Binomial Expansion for Rational n | 有理数指数 n 的展开
The true power of the binomial expansion emerges when we replace the positive integer n with any rational number n. For |x|<1 we can write:
当我们将正整数 n 替换为任意有理数 n 时，二项式展开的真正威力才显现出来。对于 |x|<1，我们可以写成：
(1+x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …
This is now an infinite series unless n is a non-negative integer. It converges only when |x|<1 (or in some special cases).
除非 n 是非负整数，否则这现在是一个无穷级数。仅当 |x|<1（或某些特殊情况）时它才收敛。
7. The Condition for Convergence |x|<1 | 收敛条件 |x|<1
The series for (1+x)ⁿ with non-integer n converges absolutely if |x|<1. The proof involves ratio tests from pure mathematics. In practice, AQA exam questions always specify the range of validity.
对于非整数 n 的 (1+x)ⁿ 级数，当 |x|<1 时绝对收敛。证明要用到纯数学中的比值审敛法。在实际考试中，AQA 试题总会明确给出的有效范围。
|x| < 1
For example, √1+x = (1+x)^½ = 1 + ½x − ⅛x² + … is valid for −1 < x < 1.
例如，√1+x = (1+x)^½ = 1 + ½x − ⅛x² + … 在 −1 < x < 1 时有效。
8. Using Partial Fractions with Binomial Expansion | 结合部分分式
When expanding a rational expression, we first decompose it into partial fractions, then apply the binomial expansion to each part individually.
当展开有理式时，我们先将其分解为部分分式，然后分别对每一部分应用二项式展开。
= 1/((1−2x)(1+x)). Write f(x) = A/(1−2x) + B/(1+x). Solving gives A = 2/3, B = 1/3. Then:
设 f(x) = 1/((1−2x)(1+x))。写出 f(x) = A/(1−2x) + B/(1+x)。解得 A = 2/3, B = 1/3。于是：
(2/3)(1−2x)⁻¹ + (1/3)(1+x)⁻¹
Each term can now be expanded using (1+u)⁻¹ = 1 − u + u² − u³ + … with an appropriate u.
现在可以分别利用 (1+u)⁻¹ = 1 − u + u² − u³ + …（取合适的 u）展开每一项。
9. Applications in Approximations | 在近似计算中的应用
The binomial expansion allows us to approximate powers and roots with great efficiency. For instance, to approximate √1.05, set x=0.05 in (1+x)^½:
二项式展开使我们能够高效地近似计算乘方与根式。例如，要近似 √1.05，在 (1+x)^½ 中令 x=0.05：
1 + ½(0.05) − ⅛(0.05)² ≈ 1.0246875
Compare with the calculator value 1.024695 – an error of less than 10⁻⁵. This method underpins many numerical algorithms.
与计算器所得 1.024695 相比，误差小于 10⁻⁵。这一方法构成了许多数值算法的基础。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
Students often forget the validity condition or misapply the general formula when n is negative or fractional. Here are crucial tips:
学生经常忘记有效性条件，或在 n 为负数或分数时错用通项公式。以下是一些关键提示：
- Always check that |x|<1 before using the infinite binomial expansion.
- Write (a+bx)ⁿ as aⁿ(1 + (b/a)x)ⁿ to use the standard form.
- For partial fractions, ensure the numerator degree is lower than the denominator.
- When finding a particular term, use the general term formula carefully with k starting at 0.
- State the range of validity explicitly in your answer.
中文翻译：始终检查是否 |x|<1 再使用无穷二项展开；将 (a+bx)ⁿ 写成 aⁿ(1 + (b/a)x)ⁿ 以利用标准形式；使用部分分式时要确保分子次数低于分母；求特定项时注意从 k=0 开始正确使用通项公式；在答案中明确写出有效范围。
In summary, the binomial expansion is a cornerstone of A-Level mathematics. Mastering both the finite and infinite versions gives you a powerful toolkit for algebra, series, and approximation. Practice the convergence conditions and the interplay with partial fractions to secure full marks in exam questions.
总之，二项式展开是 A-Level 数学的基石。掌握有限与无限两种形式，就拥有了解代数、级数和近似计算的强大工具。务必练习收敛条件以及与部分分式的结合使用，从而在考试中稳定获得满分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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