📚 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, so a = 4, b = -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。
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 x^n. The standard finite expansion is given below.
如果 n 是正整数，展开式到 x^n 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 nCr is the binomial coefficient, equal to n! / [r!(n – r)!]. On a calculator it is usually labelled nCr.
符号 nCr 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 x^0 through to x^n. The powers of a decrease from a^n down to a^0.
由于 n 是正整数，展开式恰好有 n + 1 项，x 的幂从 x^0 一直增加到 x^n；a 的幂从 a^n 一直减少到 a^0。
3. Pascal’s triangle and binomial coefficients | 帕斯卡三角与二项式系数
For small values of n, Pascal’s triangle gives a quick way to list the binomial coefficients 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。
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。
Tr+1 = nCr an−r(bx)r = nCr an−r br xr
For example, in the expansion of (2 + 5x)^8, the term in x^3 corresponds to r = 3. Its coefficient is 8C3 × 25 × 53 = 56 × 32 × 125 = 224000. The term is therefore 224000x^3.
例如，在 (2 + 5x)^8 的展开式中，x^3 项对应 r = 3。其系数为 8C3 × 25 × 53 = 56 × 32 × 125 = 224000。因此该项为 224000x^3。
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。
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。
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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