📚 The Binomial Expansion | 二项式展开
In Edexcel A-Level Mathematics, the binomial expansion is a core Pure Mathematics topic. It allows us to expand expressions of the form (a+b)n as a finite polynomial when n is a positive integer, and as an infinite series when n is a rational number. The topic connects algebra, series, approximations, and later calculus ideas, and it appears frequently in both AS and A2 assessments.
在爱德思 A-Level 数学中，二项式展开是纯数学的核心专题。当 n 为正整数时，它能把 (a+b)n 展开为有限多项式；当 n 为有理数时，它能把 (1+x)n 展开为无穷级数。该专题连接了代数、级数、近似计算以及后续的微积分思想，在 AS 和 A2 考试中频繁出现。
1. What Is a Binomial? | 什么是二项式？
A binomial is an algebraic expression that contains exactly two terms joined by addition or subtraction. Typical examples are (a+b), (2x−3), and (1+½x). When a binomial is raised to a power, direct multiplication becomes very inefficient for high powers, so we use a systematic expansion formula.
二项式是恰好包含两个项、并由加法或减法连接的代数式。典型的例子有 (a+b)、(2x−3) 和 (1+½x)。当二项式被乘方时，高次幂的直接相乘会变得非常低效，因此我们使用系统的展开公式。
2. Pascal's Triangle and nCr Notation | 帕斯卡三角与组合数记号
For small positive integer powers, Pascal's triangle gives the coefficients of (a+b)n quickly. The coefficient of the term an−rbr is the binomial coefficient nCr, where nCr = n! / [r!(n−r)!]. This is read as 'n choose r'.
对于较小的正整数幂，帕斯卡三角可以快速给出 (a+b)n 的系数。an−rbr 这一项的系数是二项式系数 nCr，其中 nCr = n! / [r!(n−r)!]，读作“n 选 r”。
y nCr = nC(n−r), so the coefficients in a binomial expansion are symmetric from both ends.
nCr 表示从 n 个对象中无序地选出 r 个对象的方法数。它满足对称性 nCr = nC(n−r)，因此二项式展开的系数从两端对称。
| n | Coefficients of (a+b)n |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
3. The Binomial Theorem for Positive Integer n | 正整数指数二项式定理
For any positive integer n, the expansion of (a+b)n is finite and can be written as:
对于任意正整数 n，(a+b)n 的展开是有限的，可以写作：
(a + b)n = nC0 an + nC1 an−1b + nC2 an−2b² + ... + nCr an−rbr + ... + nCn bn
This can also be expressed compactly using sigma notation:
这也可以用求和符号简洁地表示：
(a + b)n = Σr=0n nCr an−rbr
The powers of a decrease from n to 0, while the powers of b increase from 0 to n. In every single term, the two exponents add up to n.
a 的幂从 n 递减到 0，而 b 的幂从 0 递增到 n。在每一项中，两个指数之和都等于 n。
4. The General Term | 通项
The general term is often written as Tr+1 because the first term corresponds to r = 0. It is the term that contains br:
通项通常写作 Tr+1，因为第一项对应 r = 0。它是含有 br 的那一项：
Tr+1 = nCr an−rbr, r = 0, 1, 2, ..., n
re expansion. For example, to find the term in x³ in (2+3x)⁵, set a=2, b=3x, n=5, and r=3:
该公式在求某一特定项或系数时非常有用，无需写出整个展开式。例如，要求 (2+3x)⁵ 中 x³ 的项，令 a=2、b=3x、n=5、r=3：
<
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图