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Expanding (1 + x)^n | (1 + x)ⁿ 的展开

2026-08-30 09:44 · 本文部分内容由AI辅助创作

📚 Expanding (1 + x)n | (1 + x)ⁿ 的展开

The binomial expansion is one of the most fundamental algebraic techniques in A-Level mathematics. Understanding how to expand (1 + x)ⁿ is essential for success in Edexcel examinations, as it appears in pure mathematics, statistics, and mechanics contexts.

二项式展开是 A-Level 数学中最基础的代数技巧之一。理解如何展开 (1 + x)ⁿ 对 Edexcel 考试的成功至关重要，它出现在纯数学、统计和力学等多个领域。

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

means writing it as a sum of terms without brackets. For example, (1 + x)² = 1 + 2x + x².

二项式表达式是恰好包含两项的代数表达式，如 (1 + x)。将一个二项式提升到 n 次幂时，展开意味着将其写成不带括号的项之和。例如，(1 + x)² = 1 + 2x + x²。

The expansion follows a clear pattern that can be generalised for any power n, whether n is a positive integer, a negative integer, or a fraction.

展开遵循一个清晰的规律，可以推广到任意幂 n，无论 n 是正整数、负整数还是分数。

2. Pascal's Triangle | 帕斯卡三角

Pascal's triangle provides a visual way to find the coefficients in the expansion of (1 + x)ⁿ for small positive integer values of n. Each row begins and ends with 1, and each interior number is the sum of the two numbers above it.

帕斯卡三角提供了一种直观的方法来找到 (1 + x)ⁿ 对较小的正整数 n 展开时的系数。每一行以 1 开头和结尾，每个内部的数字是它上方两个数字之和。

For instance:
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1

例如：
第 0 行：1
第 1 行：1 1
第 2 行：1 2 1
第 3 行：1 3 3 1
第 4 行：1 4 6 4 1

to the coefficients of the expansion of (1 + x)ⁿ. Thus (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.

第 n 行中的系数直接对应 (1 + x)ⁿ 展开中的系数。因此 (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴。

3. 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:

对于正整数 n，二项式定理表述为：

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

This formula uses factorials in the denominators. The general form can also be written using binomial coefficients:

该公式在分母中使用阶乘。一般形式也可以用二项式系数来书写：

(1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + … + ⁿCₙxⁿ

Here ⁿCₖ = n! / (k!(n−k)!) and represents the number of ways to choose k items from a set of n items.

这里 ⁿCₖ = n! / (k!(n−k)!) 表示从 n 个物品中选取 k 个物品的方法数。

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4. Factorials and nCr Notation | 阶乘与组合数记号

The factorial of a positive integer n, written as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1.

正整数 n 的阶乘，记为 n!，是从 1 到 n 的所有正整数的乘积。例如，5! = 5 × 4 × 3 × 2 × 1 = 120。按照约定，0! = 1。

The binomial coefficient ⁿCₖ, also written as nCk or C(n, k), is calculated as:

二项式系数 ⁿCₖ（也可以写为 nCk 或 C(n, k)）的计算方式为：

ⁿCₖ = n! / (k!(n−k)!)

For example, ⁵C₂ = 5! / (2! × 3!) = 120 / (2 × 6) = 10. This is the coefficient of x² in the expansion of (1 + x)⁵.

例如，⁵C₂ = 5! / (2! × 3!) = 120 / (2 × 6) = 10。这就是 (1 + x)⁵ 展开式中 x² 的系数。

5. The General Term | 一般项

In the expansion of (1 + x)ⁿ where n is a positive integer, the (k+1)th term is given by:

在 (1 + x)ⁿ（n 为正整数）的展开中，第 (k+1) 项由下式给出：

Tₖ₊₁ = ⁿCₖ xᵏ

xpansion. For example, to find the term in x⁵ in the expansion of (1 + x)¹², we set k = 5:

当你需要找到特定项而不想写出整个展开式时，这个一般项非常有用。例如，要找到 (1 + x)¹² 展开中 x⁵ 的项，我们令 k = 5：

T₆ = ¹²C₅ x⁵ = 792x⁵

Some exam questions ask for the coefficient of a specific power, and using the general term formula is the most efficient approach.

有些考试题目要求特定次幂的系数，使用一般项公式是最有效的方法。

6. Expanding (1 + x)ⁿ for Positive Integer n | 正整数 n 时 (1 + x)ⁿ 的展开

When n is a positive integer, the expansion of (1 + x)ⁿ terminates after exactly n + 1 terms. This means the expansion is a finite polynomial with no approximation involved.

当 n 是正整数时，(1 + x)ⁿ 的展开恰好包含 n + 1 项后终止。这意味着展开式是一个有限多项式，不涉及任何近似。

The key properties of this expansion are:

此展开的关键性质为：