📚 8F Binomial Expansion: Preliminary Answers | 8F 二项式展开：初步答案解析
This article provides a structured walkthrough of preliminary answers for Exercise 8F on binomial expansion. It covers the core techniques for expanding (a + bx)ⁿ, finding a specific term, comparing coefficients, and checking your working. The explanations are designed to help A-level mathematics students build confidence before a formal assessment.
本文为二项式展开练习 8F 的初步答案提供结构化解析。内容涵盖展开 (a + bx)ⁿ 的核心技巧、求指定项、比较系数以及检查解题过程。这些讲解旨在帮助 A-level 数学学生在正式评估前建立信心。
1. What Is the Binomial Expansion? | 什么是二项式展开？
A binomial is an algebraic expression with exactly two terms, such as (a + b) or (1 + 2x). Binomial expansion is the process of writing a power of a binomial as a sum of terms without brackets.
二项式是恰好包含两项的代数式，例如 (a + b) 或 (1 + 2x)。二项式展开就是把二项式的乘方写成不含括号的多项式之和。
When the power n is a positive integer, the expansion has n + 1 terms. Each term involves a combination coefficient, a power of the first term, and a power of the second term. This topic is a cornerstone of A-level pure mathematics and appears frequently in Exercise 8F.
当指数 n 为正整数时，展开式共有 n + 1 项。每一项都包含一个组合系数、第一项的某个幂和第二项的某个幂。该专题是 A-level 纯数学的基石，也经常出现在练习 8F 中。
2. Key Formula for (a + bx)ⁿ | (a + bx)ⁿ 的关键公式
For a positive integer n, the binomial expansion is given by the formula below. The notation ⁿCₖ means the number of combinations of n items taken k at a time.
对于正整数 n，二项式展开由以下公式给出。记号 ⁿCₖ 表示从 n 个元素中取出 k 个的组合数。
(a + bx)ⁿ = aⁿ + n·aⁿ⁻¹·(bx) + [n(n − 1) ÷ 2]·aⁿ⁻²·(bx)² + ⋯ + (bx)ⁿ
The general combination coefficient is ⁿCₖ = n! ÷ [k!(n − k)!]. In the expanded form, the kth term uses aⁿ⁻ᵏ and (bx)ᵏ, so the coefficient of xᵏ combines ⁿCₖ with aⁿ⁻ᵏ and bᵏ.
一般组合系数为 ⁿCₖ = n! ÷ [k!(n − k)!]。在展开式中，第 k 项使用 aⁿ⁻ᵏ 和 (bx)ᵏ，因此 xᵏ 的系数由 ⁿCₖ、aⁿ⁻ᵏ 和 bᵏ 共同组成。
3. Finding the General Term | 求通项公式
The (r + 1)th term in the expansion of (a + bx)ⁿ is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ (bx)ʳ. This expression is called the general term and is essential for answering coefficient and term-finding questions.
在 (a + bx)ⁿ 的展开式中，第 (r + 1) 项为 Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ (bx)ʳ。这个表达式称为通项，是解答求系数和求指定项问题的关键。
Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ (bx)ʳ
To find the term containing a particular power of x, set the exponent of x in Tᵣ₊₁ equal to the target value, solve for r, and then substitute back. This avoids expanding the whole bracket and saves time in Exercise 8F.
要求含有特定 x 次幂的项时，先令 Tᵣ₊₁ 中 x 的指数等于目标值，解出 r，再代回通项。这样可以避免展开整个括号，在练习 8F 中节省时间。
4. Worked Example: Expanding (1 + 2x)⁵ | 例题：展开 (1 + 2x)⁵
Consider (1 + 2x)⁵. The binomial coefficients for n = 5 are 1, 5, 10, 10, 5, 1. Each term is found by raising 2x to the relevant power and multiplying by the matching binomial coefficient.
考虑 (1 + 2x)⁵。当 n = 5 时，二项式系数为 1、5、10、10、5、1。将 2x 升到相应次幂，再乘以对应的二项式系数，即可得到每一项。
(1 + 2x)⁵ = 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵
| r | Term | Coefficient |
|---|---|---|
| 0 | ⁵C₀ · 1⁵ · (2x)⁰ | 1 |
| 1 | ⁵C₁ · 1⁴ · (2x)¹ | 10x |
| 2 | ⁵C₂ · 1³ · (2x)² | 40x² |
| 3 | ⁵C₃ · 1² · (2x)³ | 80x³ |
| 4 | ⁵C₄ · 1¹ · (2x)⁴ | 80x⁴ |
| 5 | ⁵C₅ · 1⁰ · (2x)⁵ | 32x⁵ |
For example, when r = 3, we get ⁵C₃ · 1² · (2x)³ = 10 × 8x³ = 80x³. This confirms that the coefficient 80 comes from multiplying the binomial coefficient by 2³.
例如，当 r = 3 时，得到 ⁵C₃ · 1² · (2x)³ = 10 × 8x³ = 80x³。这说明了系数 80 是由二项式系数乘以 2³ 得到的。
5. Coefficient Comparison in Exam Questions | 考试题中的系数比较
Many Exercise 8F questions ask for the coefficient of x² or x³ in a given expansion. The fastest method is to use the general term, equate the power of x to the required value, and solve for r.
练习 8F 中的许多题目要求求给定展开式中 x² 或 x³ 的系数。最快的方法是使用通项，令 x 的幂等于所需值，然后解出 r。
For example, find the coefficient of x³ in (3 + 2x)⁶. The general term is ⁶Cᵣ 3⁶⁻ʳ (2x)ʳ. The power of x is r, so we need r = 3.
例如，求 (3 + 2x)⁶ 中 x³ 的系数。通项为 ⁶Cᵣ 3⁶⁻ʳ (2x)ʳ。x 的幂为 r，因此需要 r = 3。
T₄ = ⁶C₃ · 3³ · (2x)³ = 20 × 27 × 8x³ = 4320x³
Thus the coefficient of x³ is 4320. Always write the full term, including the constants, before extracting the coefficient to avoid missing factors.
因此 x³ 的系数为 4320。务必写出完整项，包括常数因子，再提取系数，以免漏掉因子。
6. Using Partial Fractions with Binomial Expansion | 二项式展开与部分分式结合
Some advanced parts of Exercise 8F involve rational expressions such as 1 ÷ [(1 + x)(1 − 2x)]. These cannot be expanded directly, so you first split the expression into partial fractions.
练习 8F 中较难的部分会涉及形如 1 ÷ [(1 + x)(1 − 2x)] 的有理式。这些式子不能直接展开，因此需要先拆成部分分式。
1 ÷ [(1 + x)(1 − 2x)] = (1/3) ÷ (1 + x) + (2/3) ÷ (1 − 2x)
Each partial fraction can then be expanded as a binomial series. For example, (1 + x)⁻¹ = 1 − x + x² − x³ + ⋯ and (1 − 2x)⁻¹ = 1 + 2x + 4x² + 8x³ + ⋯. Combine like terms to obtain the final expansion.
每个部分分式都可以用二项式级数展开。例如 (1 + x)⁻¹ = 1 − x + x² − x³ + ⋯，而 (1 − 2x)⁻¹ = 1 + 2x + 4x² + 8x³ + ⋯。合并同类项即可得到最终展开式。
7. Approximations Using Binomial Expansion | 用二项式展开做近似计算
When the value of x is small, higher powers such as x³, x⁴, and above are negligible. The binomial expansion can therefore be used to produce a polynomial approximation for a function.
当 x 的值很小时，x³、x⁴ 及更高次幂可以忽略。因此二项式展开可用来生成函数的多项式近似。
(1 + x)⁸ ≈ 1 + 8x + 28x² for small x
This is useful for estimating values such as (1.02)⁸. Put x = 0.02, so (1.02)⁸ ≈ 1 + 8(0.02) + 28(0.02)² = 1.1712. The exact value is approximately 1.1717, so the approximation is very close.
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