A-Level Mathematics: Binomial Expansion Formula and Applications | A-Level 数学：二项式展开公式与应用

📚 A-Level Mathematics: Binomial Expansion Formula and Applications | A-Level 数学：二项式展开公式与应用

The binomial expansion is one of the most essential tools in A-Level Mathematics. It lets us expand expressions of the form (a + b)ⁿ quickly and systematically, without performing long multiplications by hand. A clear understanding of the binomial theorem is required across many topics, including algebra, sequences, approximations and even statistics.

二项式展开是 A-Level 数学中最核心的工具之一。它让我们能够系统、快速地展开 (a + b)ⁿ 形式的表达式，而无需逐项手动乘法。清晰理解二项式定理是解决代数、数列、近似计算乃至统计学中许多问题的基础。

1. What Is the Binomial Theorem? | 什么是二项式定理

For a positive integer n, the binomial theorem states that (a + b)ⁿ can be written as a sum of n+1 terms. The coefficient of each term is a binomial coefficient, often written as C(n,k) or ⁿCₖ.

对于正整数 n，二项式定理指出 (a + b)ⁿ 可以写成 n+1 项之和。每一项的系数称为二项式系数，通常记作 C(n,k) 或 ⁿCₖ。

(a + b)ⁿ = Σₖ₌₀ⁿ C(n,k) aⁿ⁻ᵏ bᵏ

Here k is an integer from 0 to n. In the first term k = 0, in the last term k = n. The powers of a decrease from n to 0, while the powers of b increase from 0 to n.

其中 k 是从 0 到 n 的整数。第一项中 k = 0，最后一项中 k = n。a 的幂从 n 递减到 0，b 的幂从 0 递增到 n。

2. Factorials and Binomial Coefficients | 阶乘与二项式系数

The binomial coefficient C(n,r) counts how many ways we can choose r objects from n different objects. It is calculated using factorials.

二项式系数 C(n,r) 表示从 n 个不同物体中选取 r 个物体的方法数，它可以通过阶乘计算。

C(n,r) = n! / (r!(n − r)!)

For example, n! = n × (n−1) × (n−2) × … × 2 × 1, and 0! is defined as 1.

例如，n! = n × (n−1) × (n−2) × … × 2 × 1，并且我们定义 0! = 1。

Also, C(n,0) = C(n,n) = 1 and C(n,1) = n. A useful symmetry property is C(n,r) = C(n,n−r).

同时，C(n,0) = C(n,n) = 1，C(n,1) = n。还有一个常用的对称性质：C(n,r) = C(n,n−r)。

C(5,2) = 5! / (2! × 3!) = (5 × 4) / (2 × 1) = 10

3. The General Term | 通项公式

In the expansion of (a + b)ⁿ, the (r+1)th term is often called the general term. This is especially useful when we only need one specific term, such as the term in x³ or the constant term.

在 (a + b)ⁿ 的展开式中，第 r+1 项通常被称为通项。当我们只需要某一特定项时，例如 x³ 项或常数项，通项公式特别有用。

T(r+1) = C(n,r) aⁿ⁻ʳ bʳ

Suppose we want the term in x³ in the expansion of (x + 2)⁵. Since a = x, we need n − r = 3, so r = 2.

假设我们要求 (x + 2)⁵ 展开式中 x³ 项的系数。因为 a = x，所以需要 n − r = 3，即 r = 2。

T(3) = C(5,2) x³ × 2² = 10 × x³ × 4 = 40x³

So the coefficient of x³ is 40.

因此，x³ 的系数为 40。

4. Pascal’s Triangle | 杨辉三角

Pascal’s triangle is a visual way to list the binomial coefficients. Row n, starting from n = 0, gives the coefficients of (a + b)ⁿ.

杨辉三角是以直观方式列出二项式系数的工具。第 n 行，从 n = 0 开始，给出 (a + b)ⁿ 的展开系数。

Each number is the sum of the two numbers directly above it. This pattern matches the values of C(n,r).

每个数都等于其正上方的两个数之和。这个规律与 C(n,r) 的取值完全对应。

5. Expanding (a + b)ⁿ | 展开 (a + b)ⁿ

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）