📚 Binomial Expansion for WJEC A-Level Maths: Key Revision | A-Level WJEC 数学：二项式展开 考点精讲
Binomial expansion is a core topic in the WJEC A-Level Mathematics specification, bridging elementary algebra with more advanced series work. You must be confident expanding powers of binomials both for positive integer exponents and for negative or fractional exponents, while always keeping validity conditions in mind. This article breaks down every essential idea, from Pascal’s triangle to infinite series approximations, so you can answer any binomial question with certainty.
二项式展开是 WJEC A-Level 数学考纲中的核心内容，连接着基础代数与更高阶的级数知识。你必须熟练掌握正整数指数的展开方法，也要会处理负数指数和分数指数的情形，同时始终关注展开的有效性条件。本文拆解每个关键概念，从帕斯卡三角形到无穷级数近似，帮助你自信应对任何二项式考题。
1. What Is the Binomial Theorem? | 什么是二项式定理？
The binomial theorem gives a systematic way of expanding expressions of the form (a + b)ⁿ. When n is a positive integer, the expansion is a finite sum. For other values of n, it becomes an infinite series that converges only when certain conditions are met. In WJEC exams, you will be expected to use both the finite and the infinite form.
二项式定理提供了展开形如 (a + b)ⁿ 的表达式的系统方法。当 n 为正整数时，展开式为有限和。对于其他 n 值，展开会变成无穷级数，且只在特定条件满足时才收敛。在 WJEC 考试中，你需要熟练使用有限和无穷两种形式。
2. Pascal’s Triangle and Combinations | 帕斯卡三角形与组合数
For positive integer n, the coefficients of (a + b)ⁿ can be read directly from Pascal’s triangle. Each coefficient can also be written as a combination: the term containing aⁿ⁻ᵏ bᵏ has coefficient C(n, k) = n! / (k!(n − k)!). You must be able to use the C(n, k) notation or the nCr button on your calculator efficiently.
对于正整数 n，(a + b)ⁿ 的系数可以直接从帕斯卡三角形读出。每个系数也可以写成组合数形式：含有 aⁿ⁻ᵏ bᵏ 的项系数为 C(n, k) = n! / (k!(n − k)!)。你必须能够熟练使用 C(n, k) 记法或计算器上的 nCr 按键。
- Write C(n, 0) = 1, C(n, 1) = n, C(n, 2) = n(n−1)/2, etc. | 牢记 C(n, 0) = 1, C(n, 1) = n, C(n, 2) = n(n−1)/2 等。
- The symmetric property C(n, k) = C(n, n−k) often saves time. | 对称性质 C(n, k) = C(n, n−k) 经常能节省时间。
3. Expanding (a + b)ⁿ for Positive Integer n | 正整数指数 (a + b)ⁿ 的展开
The full expansion is: (a + b)ⁿ = Σₖ₌₀ⁿ C(n, k) aⁿ⁻ᵏ bᵏ. This means you start with k = 0 (the term aⁿ) and increase k until n. The powers of a descend while powers of b ascend, and the sum of the exponents in every term equals n.
完整展开式为：(a + b)ⁿ = Σₖ₌₀ⁿ C(n, k) aⁿ⁻ᵏ bᵏ。这意味着从 k = 0（对应项 aⁿ）开始，逐步增加 k 直至 n。a 的指数逐渐减小，b 的指数逐渐增大，每一项中两个指数之和恒等于 n。
For example, (2x + 3)⁴ = C(4,0)(2x)⁴(3)⁰ + C(4,1)(2x)³(3)¹ + C(4,2)(2x)²(3)² + C(4,3)(2x)¹(3)³ + C(4,4)(2x)⁰(3)⁴. Simplify each term carefully, especially when coefficients are large. | 例如，(2x + 3)⁴ = C(4,0)(2x)⁴(3)⁰ + C(4,1)(2x)³(3)¹ + C(4,2)(2x)²(3)² + C(4,3)(2x)¹(3)³ + C(4,4)(2x)⁰(3)⁴。化简每一项时要格外小心，尤其是系数较大时。
4. The General Term and Finding a Specific Coefficient | 一般项与求特定项的系数
WJEC often asks you to find a particular term without writing the whole expansion. The general term is Tₖ₊₁ = C(n, k) aⁿ⁻ᵏ bᵏ, where k runs from 0 to n. To find the term in xᵐ, set the exponent of x equal to m and solve for k. Then substitute back to get the coefficient.
WJEC 考试常要求你不写出全部展开式而找出某一项。一般项为 Tₖ₊₁ = C(n, k) aⁿ⁻ᵏ bᵏ，其中 k 从 0 取到 n。求含 xᵐ 的项时，令 x 的指数等于 m，解出 k，再代回求系数。
Example: Find the coefficient of x³ in (2 + x)⁷. | 示例：求 (2 + x)⁷ 中 x³ 的系数。
General term: Tₖ₊₁ = C(7, k) 2⁷⁻ᵏ xᵏ. | 一般项：Tₖ₊₁ = C(7, k) 2⁷⁻ᵏ xᵏ。
Set k = 3. Then coefficient = C(7, 3) × 2⁷⁻³ = 35 × 2⁴ = 35 × 16 = 560. | 令 k = 3，系数 = C(7, 3) × 2⁷⁻³ = 35 × 16 = 560。
5. Expanding (1 + x)ⁿ for Any Rational n | 任意有理数 n 的 (1 + x)ⁿ 展开
When n is not a positive integer, the expansion becomes an infinite series: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … This series is valid only when |x| < 1. WJEC expects you to know this form and its validity condition precisely.
当 n 不是正整数时，展开式变为无穷级数：(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … 这个级数仅在 |x| < 1 时有效。WJEC 要求你精确掌握这一形式和它的有效性条件。
Note carefully: the series continues forever. You will usually be asked to write the first four terms or to expand up to a given power of x. | 特别注意：级数无限延续。题目通常要求写出前四项或展开到 x 的某次幂为止。
6. Expansions with Negative and Fractional Powers | 负数与分数指数的展开
For n = −1, the expansion of (1 + x)⁻¹ gives 1 − x + x² − x³ + … valid for |x| < 1. Similarly, (1 + x)⁻² = 1 − 2x + 3x² − 4x³ + … For fractional powers such as n = ½, you obtain 1 + ½x − ⅛x² + 1/16 x³ − … In exam questions, you must handle rational n with the same formula, carefully computing each coefficient.
当 n = −1 时，(1 + x)⁻¹ 的展开为 1 − x + x² − x³ + …，有效范围 |x| < 1。类似地，(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + …；对于分数指数，如 n = ½，得到 1 + ½x − ⅛x² + 1/16 x³ − …。考试中，你必须用同一公式处理有理数 n，仔细计算每一个系数。
Work step-by-step: identify n, write the general factor n(n−1)(n−2)…/(k!), and avoid arithmetic slip-ups with signs and fractions. | 逐步运算：先确定 n，写出一般因子 n(n−1)(n−2)…/(k!)，并注意符号和分数运算，避免低级错误。
7. Validity Conditions for Infinite Series | 无穷级数的有效性条件
For (1 + u)ⁿ where n is not a positive integer, the expansion is valid ONLY when |u| < 1. If the expression is of the form (a + bx)ⁿ, you must first rewrite it as aⁿ (1 + (b/a)x)ⁿ, then the condition becomes |(b/a)x| < 1, i.e., |x| < |a/b|. Never forget to state the validity range in your answer.
对非正整数 n 的 (1 + u)ⁿ 展开，仅在 |u| < 1 时有效。如果表达式形如 (a + bx)ⁿ，必须首先改写为 aⁿ (1 + (b/a)x)ⁿ，然后条件变为 |(b/a)x| < 1，即 |x| < |a/b|。永远不要忘记在答案中注明有效范围。
For example, expand (4 − x)⁻² up to x³ and state when it is valid. First factor out 4: [4(1 − x/4)]⁻² = 4⁻² (1 − x/4)⁻² = 1/16 × (1 − x/4)⁻². Then expand using n = −2 and u = −x/4. Condition: |−x/4| < 1 → |x| < 4. | 例如，展开 (4 − x)⁻² 至 x³ 并说明有效范围。先提取因子 4：[4(1 − x/4)]⁻² = 4⁻² (1 − x/4)⁻² = 1/16 × (1 − x/4)⁻²，然后用 n = −2, u = −x/4 展开。条件：|−x/4| < 1 → |x| < 4。
8. Expanding Expressions of the Form (a + bx)ⁿ | 展开形如 (a + bx)ⁿ 的表达式
When the binomial is not simply (1 + x), you must factor out a to create the standard form. Steps: (a + bx)ⁿ = aⁿ (1 + (b/a)x)ⁿ if a ≠ 0. Then apply the series expansion, and multiply every term by aⁿ at the end or keep it outside. This method works for any rational n.
当二项式不是简单的 (1 + x) 时，必须提取因子 a 构造成标准形式。步骤：(a + bx)ⁿ = aⁿ (1 + (b/a)x)ⁿ，其中 a ≠ 0。然后套用级数展开，最后再乘上 aⁿ 或者保留在外面。这种方法适用于任何有理数 n。
Example: Expand √(9 + 2x) as a series up to x². | 示例：将 √(9 + 2x) 展开为级数至 x²。
Rewrite as (9 + 2x)½ = 9½ (1 + (2x/9))½ = 3 (1 + 2x/9)½. | 改写为 (9 + 2x)½ = 9½ (1 + 2x/9)½ = 3 (1 + 2x/9)½。
Use n = ½: (1 + u)½ ≈ 1 + ½u − ⅛u². Substitute u = 2x/9, then multiply by 3. | 用 n = ½：(1 + u)½ ≈ 1 + ½u − ⅛u²，代入 u = 2x/9，最后乘以 3。
Result: 3 [1 + (1/2)(2x/9) − (1/8)(4x²/81) + …] = 3 + x/3 − x²/54 + … Valid for |2x/9| < 1 → |x| < 9/2. | 结果：3 [1 + x/9 − x²/162 + …] = 3 + x/3 − x²/54 + …，有效范围 |2x/9| < 1 → |x| < 9/2。
9. Using Binomial Expansion for Approximations | 用二项式展开求近似值
You can approximate values like √(1.02) or 1/(0.98)³ by writing them in binomial form and taking the first few terms. For instance, √(1.02) = (1 + 0.02)½ ≈ 1 + ½(0.02) − ⅛(0.02)² + … Always check that the x value falls within the validity interval.
你可以把 √(1.02) 或 1/(0.98)³ 这样的值写成二项式形式，并取前几项进行近似。例如 √(1.02) = (1 + 0.02)½ ≈ 1 + ½(0.02) − ⅛(0.02)² + …。务必检查 x 的值是否落在有效性区间内。
For higher accuracy, determine whether the next term is small enough to ignore. In a WJEC context, you might be asked to give an approximation to a specified number of decimal places and compare with the exact value.
为了更高的精确度，要判断下一项是否足够小到可以忽略。在 WJEC 题目中，可能会要求你给出精确到指定位数的小数近似值，并与真实值比较。
10. Combining Expansions and Partial Fractions | 与部分分式结合的展开
A favourite WJEC extension is to express a rational function in partial fractions and then expand each fraction binomially. For example, expand 3x/((1−x)(2+x)) by splitting into A/(1−x) + B/(2+x), rewriting each as a binomial expansion, and adding the series. This tests your algebraic manipulation and understanding of validity ranges.
WJEC 常见的拓展题型是先将有理函数分成部分分式，再对每个分式进行二项式展开。例如，展开 3x/((1−x)(2+x))，先拆成 A/(1−x) + B/(2+x)，各自改写为二项式级数，然后相加。这类题考查你的代数操作能力和对有效范围的理解。
Always find the overall validity interval by taking the most restrictive of the individual conditions. The final series will only be valid where all parts converge.
一定要通过取出各个条件中最严苛的一个来确定整体的有效区间。最终级数只在各部分都收敛的区域内有效。
11. Pitfalls and Common Mistakes | 陷阱与常见错误
• Forgetting the validity condition or stating it incorrectly is one of the biggest mark-losers. Always write |x| < something. | 忘记有效条件或写错条件是最大的失分点之一。务必写出 |x| < 某值。
• Mistaking the formula for (1 + x)ⁿ when n is not an integer. The signs and coefficients follow a strict pattern; do not treat it like (a + b)ⁿ with integer n. | 当 n 不是整数时，误用公式。符号和系数遵循严格的规律，不要把它当成整数 n 的 (a + b)ⁿ 来处理。
• Arithmetic when simplifying coefficients, especially with negative n − 1, n − 2 etc. Work slowly, write each factorial explicitly, and cancel early. | 化简系数时，尤其是在出现负的 n − 1、n − 2 等时，容易犯运算错误。慢慢计算，清楚地写出每个阶乘形式，及早约分。
• Forgetting to multiply back by aⁿ after factoring out a in (a + bx)ⁿ. | 在 (a + bx)ⁿ 中提取因子 a 后，忘记乘回 aⁿ。
12. Summary of Key Points | 考点总结
Master binomial expansion by knowing two main cases: positive integer n gives a finite sum with combinations; any rational n gives an infinite series in ascending powers of x, valid for |x| < 1 after converting to standard form. Practise finding specific terms, handling (a + bx)ⁿ, and combining with partial fractions. When approximating, always check validity and quote the condition explicitly. A methodical approach and attention to detail will secure full marks on this topic.
掌握二项式展开，关键在于两种情况：正整数 n 时是有限和，用组合数表示；任意有理数 n 时是 x 的升幂无穷级数，化成标准形式后有效范围为 |x| < 1。多练习求特定项、处理 (a + bx)ⁿ 以及与部分分式结合的题型。求近似值时，务必检验有效性并明确给出条件。有条不紊的方法和对细节的关注能帮你在这个考点上稳拿满分。
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