📚 Investigation 2 - The binomial expansion | 二项式展开探究
In this investigation, we explore the binomial expansion, a fundamental algebraic technique that allows us to expand expressions of the form (a + b)ⁿ for positive integer n, and even for rational exponents. We will discover patterns, connect them to Pascal's triangle, define binomial coefficients using combinations, and prove the binomial theorem using mathematical induction. This investigation is designed to deepen your understanding of how algebraic expansions work and how they can be applied in problem solving, approximation, and beyond.
在本次探究中，我们将探索二项式展开，这是一种基本的代数方法，可以将形如 (a + b)ⁿ 的式子展开，其中 n 为正整数，甚至可推广到有理指数。我们将发现规律，与帕斯卡三角形建立联系，通过组合数定义二项式系数，并用数学归纳法证明二项式定理。本探究旨在加深你对代数展开原理的理解，以及如何在解题、近似计算等方面应用它。
1. Introduction: What is Binomial Expansion? | 什么是二项式展开？
or higher powers, performing the multiplication directly becomes tedious. Through this investigation, we aim to find a systematic method to expand (a + b)ⁿ for any positive integer n, and later explore extensions to negative and fractional powers.
二项式是包含两项的代数式，例如 (x + y) 或 (2a - 3b)。二项式展开就是将二项式的幂乘开的过程，例如 (x + y)² = x² + 2xy + y² 或 (a + b)³ = a³ + 3a²b + 3ab² + b³。对于更高次幂，直接乘法十分繁琐。通过本次探究，我们旨在找到一种系统的方法，对任意正整数 n 展开 (a + b)ⁿ，随后还将探索负指数与分数指数幂的推广。
By investigating the coefficients of the expanded forms, we can spot patterns that lead to Pascal's triangle and the concept of binomial coefficients.
通过研究展开式中的系数，我们可以发现规律，进而引出帕斯卡三角形和二项式系数的概念。
2. Exploring Patterns: Multiplying (a+b)ⁿ by Hand | 探索规律：手动展开 (a+b)ⁿ
Let' ² = a² + 2ab + b²; (a+b)³ = a³ + 3a²b + 3ab² + b³; (a+b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.
我们首先手动计算 n = 0, 1, 2, 3, 4 时的展开式：(a+b)⁰ = 1；(a+b)¹ = a + b；(a+b)² = a² + 2ab + b²；(a+b)³ = a³ + 3a²b + 3ab² + b³；(a+b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴。
; n=3: 1, 3, 3, 1; n=4: 1, 4, 6, 4, 1. This triangular array of numbers is known as Pascal's triangle.
如果我们只看数字系数（暂时忽略 a 和 b），可得到：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。这个三角形数列就是著名的帕斯卡三角形。
Notice the symmetry and the fact that each coefficient is the sum of the two above it in the previous row. This suggests a recursive way to generate binomial coefficients.
可以看到系数具有对称性，且每个系数等于上一行相邻两个系数之和。这提示了一种递归生成二项式系数的方式。
3. Pascal's Triangle and Its Connection | 帕斯卡三角形及其联系
Pascal' 0) contains the coefficients of (a + b)ⁿ. For example, row 5 is 1, 5, 10, 10, 5, 1, which correspond to the expansion (a
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