Binomial Expansion: (a+bx)^n | 二项式展开：(a+bx)^n

📚 Binomial Expansion: (a + bx)ⁿ | 二项式展开：(a + bx)ⁿ

A binomial expression is an algebraic expression that contains exactly two terms, such as a + bx. The binomial expansion is a systematic way to write (a + bx)ⁿ as a sum of terms involving powers of x, each multiplied by a numerical coefficient.

二项式是恰好包含两项的代数式，例如 a + bx。二项式展开是将 (a + bx)ⁿ 系统地写成按 x 的幂排列的若干项之和，每一项都乘以一个数值系数。

For a positive integer n, this expansion is finite and contains exactly n + 1 terms. For negative or fractional n, the expansion becomes an infinite series, and it is valid only for certain values of x.

当 n 为正整数时，展开式是有限的，恰好有 n + 1 项。当 n 为负数或分数时，展开式变为无穷级数，并且只对某些 x 值成立。

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

The binomial coefficient C(n,r), sometimes written as ⁿCᵣ, counts the number of ways to choose r items from n items. It is defined using factorials:

二项式系数 C(n,r)（有时写作 ⁿCᵣ）表示从 n 个元素中选取 r 个元素的方法数，它通过阶乘定义：

C(n,r) = ⁿCᵣ = n! / [r!(n − r)!]

Here n! means n × (n − 1) × (n − 2) × … × 2 × 1, and by convention 0! = 1.

这里 n! 表示 n × (n − 1) × (n − 2) × … × 2 × 1，并且规定 0! = 1。

For example, C(5,2) = 5! / (2! × 3!) = 120 / (2 × 6) = 10. These coefficients also appear in Pascal’s triangle.

例如，C(5,2) = 5! / (2! × 3!) = 120 / (2 × 6) = 10。这些系数也出现在杨辉三角中。

A useful symmetry is C(n,r) = C(n,n − r). You will often use this to save time when r is close to n.

一个有用的对称性质是 C(n,r) = C(n,n − r)。当 r 接近 n 时，利用这个性质可以节省时间。

2. The General Expansion for Positive Integer n | 正整数 n 的展开通式

For a positive integer n, the binomial expansion of (a + bx)ⁿ is:

对于正整数 n，(a + bx)ⁿ 的二项式展开为：

(a + bx)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹(bx) + ⁿC₂ aⁿ⁻²(bx)² + … + ⁿCₙ (bx)ⁿ

Equivalently, the term containing (bx)ʳ is ⁿCᵣ aⁿ⁻ʳ (bx)ʳ for r = 0, 1, 2, …, n.

等价地，含有 (bx)ʳ 的项为 ⁿCᵣ aⁿ⁻ʳ (bx)ʳ，其中 r = 0, 1, 2, …, n。

Notice that the powers of a decrease from n to 0, while the powers of bx increase from 0 to n. The sum of the exponents in each term is always n.

注意 a 的幂从 n 递减到 0，而 bx 的幂从 0 递增到 n。每一项中两个指数的和始终为 n。

Because the expansion is finite, there is no restriction on x when n is a positive integer.

因为展开是有限的，所以当 n 为正整数时，对 x 没有限制。

3. Worked Examples for Positive Integer n | 正整数 n 的实例

Example 1: Expand (1 + 2x)⁴.

例 1：展开 (1 + 2x)⁴。

Using a = 1, b = 2 and n = 4, the coefficients are 1, 4, 6, 4, 1:

取 a = 1，b = 2，n = 4，系数为 1, 4, 6, 4, 1：

(1 + 2x)⁴ = 1 + 8x + 24x² + 32x³ + 16x⁴

Check: 4 × 1³ × (2x) = 8x, 6 × 1² × (2x)² = 24x², 4 × 1 × (2x)³ = 32x³, and (2x)⁴ = 16x⁴.

验算：4 × 1³ × (2x) = 8x，6 × 1² × (2x)² = 24x²，4 × 1 × (2x)³ = 32x³，(2x)⁴ = 16x⁴。

Example 2: Expand (2 − 3x)⁴.

例 2：展开 (2 − 3

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