Expanding (a + bx)^n | (a + bx)^n 的展开

📚 Expanding (a + bx)ⁿ | (a + bx)ⁿ 的展开

Binomial expansion is one of the most frequently examined topics in Edexcel A-Level Mathematics. You need to be able to expand expressions of the form (a + bx)ⁿ for positive integer powers, where the expansion is finite, and for fractional or negative powers, where the expansion is an infinite series that only converges for a limited range of x.

二项式展开是 Edexcel A-Level 数学中最常考查的主题之一。你需要能够展开 (a + bx)ⁿ 形式的表达式：当 n 为正整数时，展开式是有限的；当 n 为分数或负数时，展开式是无限级数，并且只在有限的 x 范围内收敛。

1. Recognising the form (a + bx)ⁿ | 识别 (a + bx)ⁿ 形式

Always start by identifying the values of a, b and n. In Edexcel questions the bracket is often something like (2 + 3x)⁵, (4 - x/2)⁷ or (1 + 5x)^-2. If there is a subtraction, rewrite it as an addition so that the bracket becomes (a + bx) with a and b clearly defined.

开始时一定要先识别 a、b 和 n 的值。在 Edexcel 考题中，括号通常是 (2 + 3x)⁵、(4 - x/2)⁷ 或 (1 + 5x)^-2 这样的形式。如果有减法，要把它改写为加法，使括号变成 (a + bx) 的形式，并清楚地定义 a 和 b。

(a + bx)ⁿ

For example, (4 - x/2)⁷ should be treated as (4 + (-1/2)x)⁷ -1/2 and n = 7. This step prevents sign errors later when finding individual coefficients.

例如，(4 - x/2)⁷ 应视为 (4 + (-1/2)x)⁷，因此 a = 4，b = -1/2，n = 7。这一步可以防止后面求单项系数时出现符号错误。

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2. The binomial theorem for positive integer n | 正整数 n 的二项式定理

If n is a positive integer, the expansion stops at the term in xⁿ. The standard finite expansion is given below.

如果 n 是正整数，展开式到 xⁿ 项为止。标准的有限展开式如下。

(a + bx)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹(bx) + ⁿC₂ aⁿ⁻²(bx)² + ... + (bx)ⁿ

The symbol ⁿC_r is the binomial coefficient, equal to n! / [r!(n - r)!]. On a calculator it is usually labelled nCr.

符号 ⁿC_r 是二项式系数，等于 n! / [r!(n - r)!]。在计算器上通常标为 nCr。

Because n is a positive integer, there are exactly n + 1 terms in the expansion, and the powers of x increase from x⁰ through to xⁿ. The powers of a decrease from aⁿ down to a⁰.

由于 n 是正整数，展开式恰好有 n + 1 项，x 的幂从 x⁰ 一直增加到 xⁿ；a 的幂从 aⁿ 一直减少到 a⁰。

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3. Pascal's triangle and binomial coefficients | 帕斯卡三角与二项式系数

For small values of n, Pascal' without using the nCr formula. Each row corresponds to a value of n, starting from n = 0.

当 n 较小时，帕斯卡三角可以快速列出二项式系数，而不必使用 nCr 公式。每一行对应一个 n 值，从 n = 0 开始。

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
n = 5	1 5 10 10 5 1

For example, to expand (1 + 2x)⁵, the coefficients are 1, 5, 10, 10, 5, 1. The expansion is then built term by term with increasing powers of 2x.

例如，要展开 (1 + 2x)⁵，系数为 1、5、10、10、5、1。然后按照 2x 的升幂逐项写出展开式。

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4. The general term and finding one coefficient | 一般项与求单个系数

The (r + 1)th term is extremely useful when a question asks for just one specific coefficient. Instead of writing out the whole expansion, you can use the formula below.

第 r + 1 项在题目只要求某一个特定系数时非常有用。你可以使用下面的公式，而不必写出整个展开式。

T_r+1 = ⁿC_r a^n−r(bx)^r = ⁿC_r a^n−r b^r x^r

For example, in the expansion of (2 + 5x)⁸, the term in x³ corresponds to r = 3. Its coefficient is ⁸C₃ × 2⁵ × 5³ = 56 × 32 × 125 = 224000. The term is therefore 224000x³.

例如，在 (2 + 5x)⁸ 的展开式中，x³ 项对应 r = 3。其系数为 ⁸C₃ × 2⁵ × 5³ = 56 × 32 × 125 = 224000。因此该项为 224000x³。

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5. Handling a ≠ 1 | 处理 a ≠ 1 的情况

For the infinite binomial series used with fractional or negative powers, the bracket must begin with 1. If a is not 1, factorise a out of the bracket before expanding.

对于分数次幂或负数次幂使用的无限二项级数，括号内必须以 1 开头。如果 a 不等于 1，展开前需要先把 a 从括号中提取出来。

(a + bx)ⁿ = aⁿ (1 + (b/a)x)ⁿ

The range of validity is then found from the condition |(b/a)x| < 1. This gives |x| < |a/b|, assuming b ≠ 0.

然后根据条件 |(b/a)x| < 1 确定有效范围。该条件给出 |x| < |a/b|，假设 b ≠ 0。

For example, (3 + 4x)^-2 becomes 3^-2 (1 + (4/3)x)^-2 = (1/9)(1 + (4/3)x)^-2. The expansion is valid for |x| < 3/4.

例如，(3 + 4x)^-2 可写成 3^-2 (1 + (4/3)x)^-2 = (1/9)(1 + (4/3)x)^-2。展开式对 |x| < 3/4 有效。

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6. The general binomial theorem for rational n | 有理数 n 的一般二项式定理

When n is fractional or negative, the expansion does not terminate. It becomes an infinite series given by the formula below, provided |x| < 1.

当 n 是分数或负数时，展开式不会终止。它变成一个无限级数，如下式所示，但前提是 |x| < 1。

(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ... for |x| < 1

This is tested heavily in Edexcel A-Level Pure Mathematics Year 2. You must be able to write down the first few terms quickly and state the interval of validity.

这是 Edexcel A-Level 纯数学第二年考试中的重点内容。你必须能够快速写出前几项，并说明有效区间。

The coefficients follow a pattern: multiply by n, then n - 1, then n - 2, and divide by 1, 2, 3 respectively. The series continues forever unless n is a positive integer.

系数的规律是：分别乘以 n、n - 1、n - 2，再分别除以 1、2、3。除非 n 是正整数，否则级数会一直延续下去。

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7. Validity and interval of convergence | 有效性与收敛区间

Every infinite binomial expansion has a validity condition that must be stated in the answer. The expansion of (1 + y)ⁿ is valid only for |y| < 1.

每个无限二项展开式都有一个必须在答案中说明的有效性条件。(1 + y)ⁿ 的展开式仅在 |y| < 1 时有效。

For (a + bx)ⁿ, after factorising, y = (b/a)x. Therefore the condition is |(b/a)x| < 1, which simplifies to |x| < |a/b|.

对于 (a + bx)ⁿ，提取公因式后 y = (b/a)x。因此条件是 |(b/a)x| < 1，化简为 |x| < |a/b|。

Example: expand (2 + 3x)^-1. Write it as 2^-1 (1 + (3/2)x)^-1. The expansion is valid for |(3/2)x| < 1, so |x| < 2/3.

例如：展开 (2 + 3x)^

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