IB WJEC Mathematics: Binomial Expansion – Key Points Revision | IB WJEC 数学：二项式展开考点精讲

📚 IB WJEC Mathematics: Binomial Expansion – Key Points Revision | IB WJEC 数学：二项式展开考点精讲

Welcome to this focused revision guide on Binomial Expansion, designed specifically for the IB and WJEC Mathematics curriculum. We will walk through the fundamental principles, tackle exam-style problems, and highlight the common pitfalls students encounter. Whether you are aiming for a solid grade in SL or diving into the deeper rational-power expansions in HL, mastering this topic is essential for algebraic fluency and calculus applications.

欢迎阅读这篇专门为 IB 和 WJEC 数学课程定制的二项式展开考点精讲。我们将从头梳理基本原理，攻克考试型难题，并指出学生最容易踩的坑。无论你的目标是 SL 的扎实分数，还是想在 HL 中深入有理数次幂的展开，吃透这个主题对提升代数运算能力和后续微积分应用都至关重要。

1. Introduction to Binomial Expansion | 二项式展开简介

A binomial is simply an algebraic expression containing two terms, such as (x + y) or (2a – 3b). Raising a binomial to a positive integer power n yields a sum of terms, and the Binomial Theorem provides a systematic way to write that sum without multiplying repeatedly.

二项式就是包含两项的代数式，例如 (x + y) 或 (2a – 3b)。将一个二项式进行正整数次幂 n 的运算会产生若干项的和，而二项式定理给出了无需反复乘开就能系统写出该和的方法。

For small values of n, many students rely on Pascal’s Triangle, but for larger n or for proofs, the theorem using combinations is indispensable.

对于较小的 n，许多同学依赖帕斯卡三角形，但当 n 较大或进行证明时，使用组合数的定理形式不可或缺。

2. Pascal’s Triangle and Combination Coefficients | 帕斯卡三角形与组合数系数

Pascal’s Triangle provides a quick way to read the coefficients for expansions of (a + b)ⁿ when n is a small positive integer. Each entry is the sum of the two entries directly above it.

帕斯卡三角形为 (a + b)ⁿ 展开提供了快速读取系数的方法，前提是 n 为较小的正整数。三角形中的每个数都是其上方两数之和。

Mathematically, these coefficients are given by the binomial coefficient ⁿC_r or C(n, r) = n! / (r! (n – r)!). They count the number of ways to choose r items from n items.

这些系数的数学本质是二项式系数 ⁿC_r 或 C(n, r) = n! / (r! (n – r)!)。它表示从 n 个物体中选出 r 个的组合数。

Example: For n = 4, the row in Pascal’s Triangle is 1, 4, 6, 4, 1, which correspond to ⁴C₀, ⁴C₁, ⁴C₂, ⁴C₃, ⁴C₄.

例如：当 n=4 时，帕斯卡三角形的行是 1, 4, 6, 4, 1，恰好对应 ⁴C₀, ⁴C₁, ⁴C₂, ⁴C₃, ⁴C₄。

3. The Binomial Theorem for Positive Integer n | 正整数次幂的二项式定理

The full binomial expansion for (a + b)ⁿ when n is a positive integer is:

(a + b)ⁿ = Σ_r=0ⁿ ⁿC_r a^n−r b^r

当 n 为正整数时，(a + b)ⁿ 的完整二项式展开为：

(a + b)ⁿ = Σ_r=0ⁿ ⁿC_r a^n−r b^r

This means the expansion has n+1 terms, starting with aⁿb⁰ and ending with a⁰bⁿ. The general term is often denoted T_r+1 = ⁿC_r a^n−r b^r.

这意味着展开式有 n+1 项，起始于 aⁿb⁰，终止于 a⁰bⁿ。通常将通项记为 T_r+1 = ⁿC_r a^n−r b^r。

Care must be taken when the binomial contains negative signs or coefficients. For (x − 2y)⁵, treat a = x and b = −2y. The sign alternates because powers of (−2y) produce negative terms when r is odd.

当二项式含有负号或系数时必须小心处理。对于 (x − 2y)⁵，可令 a = x，b = −2y。由于 (−2y) 的奇次幂产生负项，展开式的正负号会交替出现。

4. Expanding Expressions of the Form (ax ± by)ⁿ | 形如 (ax ± by)ⁿ 的展开

In practice, IB and WJEC questions often ask you to expand something like (2x + 3)⁴ or (5 − 2x)³. Always identify a and b, and apply the general term systematically.

在实际考试中，IB 和 WJEC 经常要求展开 (2x + 3)⁴ 或 (5 − 2x)³ 这样的表达式。务必明确 a 和 b ，然后系统地套用通项公式。

For (2x + 3)⁴: a = 2x, b = 3. The term T_r+1 = ⁴C_r (2x)^4−r (3)^r. Write the full expansion by evaluating for r = 0, 1, 2, 3, 4. Remember to simplify the constants and powers of x.

以 (2x + 3)⁴ 为例：a = 2x, b = 3。通项 T_r+1 = ⁴C_r (2x)^4−r (3)^r。对 r = 0, 1, 2, 3, 4 分别求值，并别忘了化简常数和 x 的幂。

A common mistake is forgetting to raise the coefficient of x to the power n−r. Expanding (2x)³ gives 8x³, not 2x³.

一个常见错误是忘记将 x 的系数也进行 (n−r) 次乘方。比如 (2x)³ 应该是 8x³，而非 2x³。

5. Finding a Specific Term or Coefficient | 求特定项或系数

One of the most examined skills is determining a specific term, often the x^k term, without writing the entire expansion. Use the general term T_r+1 and set the power of the variable equal to the required exponent.

考试中最核心的技能之一，就是在不写出全部展开式的情况下求特定项（通常是某个 x^k 项）。可以利用通项 T_r+1，将变量的指数设定为题目所要求的值。

Example: Find the coefficient of x⁶ in the expansion of (x² + 2/x)⁹. Write T_r+1 = ⁹C_r (x²)^9−r (2x⁻¹)^r. The power of x is 2(9−r) + (−1)r = 18 − 3r. Set 18 − 3r = 6 → r = 4. Then the coefficient is ⁹C₄ × 2⁴.

例如：求 (x² + 2/x)⁹ 展开式中 x⁶ 的系数。写出通项 T_r+1 = ⁹C_r (x²)^9−r (2x⁻¹)^r。x 的指数为 2(9−r) + (−1)r = 18 − 3r。令 18 − 3r = 6 得 r = 4。于是系数为 ⁹C₄ × 2⁴。

Be careful: the coefficient is the constant multiplying the variable part, so after finding r, substitute into the constant part of the general term only, excluding the variable.

注意：系数是指乘以变量部分的常数，因此求出 r 后只需代入通项的常数部分，不要包含变量。

6. Properties of Binomial Coefficients and Symmetry | 二项式系数的性质与对称性

Binomial coefficients satisfy ⁿC_r = ⁿC_n−r, which explains the symmetry in Pascal’s Triangle. They also sum to 2ⁿ, meaning Σ_r=0ⁿ ⁿC_r = 2ⁿ.

二项式系数满足 ⁿC_r = ⁿC_n−r，这解释了帕斯卡三角形的对称性。它们的总和为 2ⁿ，即 Σ_r=0ⁿ ⁿC_r = 2ⁿ。

This property is often tested in proving identities or in combinatorics questions. For instance, evaluating (1.01)⁴ using a binomial expansion can be quickly approximated using early terms.

这个性质常在证明恒等式或组合题中考查。例如，用二项式展开估算 (1.01)⁴ 可以通过前几项快速得到近似值。

Also note the recursive relation: ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r, which is the rule that generates Pascal’s Triangle.

还需掌握递推关系：ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r，这正是生成帕斯卡三角形的规则。

7. Expansion of (1 + x)ⁿ for Rational n | (1 + x)ⁿ 当 n 为有理数时的展开

When n is not a positive integer (e.g., negative or fractional), the binomial series becomes infinite. The expansion is valid only when |x| < 1.

当 n 不是正整数（例如负数或分数）时，二项式级数变成无穷级数。该展开只有在 |x| < 1 时才成立。

(1 + x)ⁿ = 1 + nx + n(n−1)/2! · x² + n(n−1)(n−2)/3! · x³ + …

This formula is provided in the IB data booklet and is essential for WJEC A-level exams. The general term is [n(n−1)…(n−r+1) / r!] x^r.

该公式在 IB 公式表中有提供，也是 WJEC A-level 考试的必备内容。通项为 [n(n−1)…(n−r+1) / r!] x^r。

Example: Expand √(1 + 2x) up to x³. Here n = 1/2, and x is replaced by 2x. The expansion is 1 + (1/2)(2x) + (1/2)(−1/2)/2! (2x)² + (1/2)(−1/2)(−3/2)/3! (2x)³ + …, simplify carefully.

例如：将 √(1 + 2x) 展开至 x³ 项。这里 n = 1/2，并把 x 替换为 2x。展开为 1 + (1/2)(2x) + (1/2)(−1/2)/2! (2x)² + (1/2)(−1/2)(−3/2)/3! (2x)³ + …，需仔细化简。

8. Validity Conditions for Infinitely Many Terms | 无穷级数的收敛条件

For the infinite series expansion of (1 + x)ⁿ to be valid, the modulus of x must be less than 1, i.e., |x| < 1. If the binomial is (a + bx)ⁿ, first factor out aⁿ to reach the form aⁿ (1 + (b/a)x)ⁿ; the requirement becomes |(b/a)x| < 1.

对于 (1 + x)ⁿ 的无穷级数展开，必须满足 x 的模小于 1，即 |x| < 1。如果二项式是 (a + bx)ⁿ，应先提取因子 aⁿ 变成 aⁿ (1 + (b/a)x)ⁿ；此时收敛条件变为 |(b/a)x| < 1。

This is a classic exam trap: asking for the expansion and then the range of x for which it is valid. Always state the condition clearly, e.g., “the expansion is valid for |3x/2| < 1 ⇒ |x| < 2/3".

这是经典的考试陷阱：题目要求写出展开式，接着还要求写出有效的 x 取值范围。一定要清晰陈述条件，例如“该展开在 |3x/2| < 1 即 |x| < 2/3 时有效。”

9. Using Binomial Expansion for Approximations | 利用二项式展开进行近似计算

When the power series is truncated after a few terms, it gives a good approximation for small x. For example, (1 + x)^1/2 ≈ 1 + x/2 − x²/8 can be used to estimate √1.02 by setting x = 0.02.

当幂级数只取前几项时，它对小量 x 能给出良好的近似。例如 (1 + x)^1/2 ≈ 1 + x/2 − x²/8，可设 x = 0.02 来估算 √1.02。

This technique often appears in questions where you are required to calculate an approximate value to a specified number of decimal places. Ensure you show the substitution step clearly and state the reason for ignoring higher-order terms.

这种技巧常出现在要求计算到指定小数位数的近似值题目中。务必清晰展示代入步骤，并说明忽略高阶项的原因。

Also be prepared to combine terms when the expression is not exactly (1 + something); for instance, 1/∛(1 − 2x) can be written as (1 − 2x)^−1/3 and expanded.

也要准备好处理表达式的变形，例如 1/∛(1 − 2x) 可写为 (1 − 2x)^−1/3 然后展开。

10. Typical Exam-Style Problems | 典型考试题型分析

Exam questions often mix positive-integer and rational-power expansions. One common structure: Part (a) asks for the expansion of (1 + 3x)⁴ using the binomial theorem. Part (b) asks to find the coefficient of x² in the expansion of (1 + 3x)⁴(2 − x)⁻² by combining series.

考试题经常混合正整数次幂和有理数次幂的展开。常见结构：第 (a) 部分用二项式定理展开 (1 + 3x)⁴；第 (b) 部分要求通过级数乘法求 (1 + 3x)⁴(2 − x)⁻² 展开式中 x² 的系数。

For combining expansions, expand each up to the required power, then multiply and collect like terms. For (2 − x)⁻², first write as 2⁻²(1 − x/2)⁻² = 1/4 (1 + 2(x/2) + 3(x/2)² + …) using the general binomial expansion with n = −2.

针对组合展开，先分别展开到所需的幂，然后相乘并合并同类项。对于 (2 − x)⁻²，可先写成 2⁻²(1 − x/2)⁻² = 1/4 (1 + 2(x/2) + 3(x/2)² + …)，这里用到 n = −2 的通项展开。

Another tricky area: using the expansion to find the coefficient of x^r from a product like (ax + b)ⁿ(cx + d)^m. Identify the pairs of terms whose powers sum to r, compute each contribution, and sum them.

另一个棘手的地方：利用展开式求形如 (ax + b)ⁿ(cx + d)^m 的乘积中 x^r 的系数。要找出幂次之和为 r 的两项组合，分别计算贡献再相加。

11. Common Mistakes and How to Avoid Them | 常见错误及规避方法

Forgetting the exponents on coefficients: In (3x)⁴, students often write 3x⁴ instead of 81x⁴. Always calculate (coefficient)^power first.

忽略系数的指数：在 (3x)⁴ 中，学生常写成 3x⁴ 而非 81x⁴。一定要先算 (系数)^幂。
Misusing the general term index: Remember T_r+1 uses r, not r−1. Many answers lose marks by starting the sum at r = 1 incorrectly.

通项下标使用错误：记住 T_r+1 使用的是 r，而非 r−1。很多答卷因错误地从 r=1 开始求和而失分。
Sign errors with negative b: Expand (a − b)ⁿ carefully; the sign is (−1)^r. Write the general term with the minus sign inside the b^r factor.

括号内为负时的符号错误：展开 (a − b)ⁿ 要格外小心；符号由 (−1)^r 决定。把负号直接放在 b^r 的因子中写出通项。
Ignoring validity for infinite series: A fully correct rational-power expansion without the validity statement often costs a mark.

忽略无穷级数的收敛条件：有理数次幂展开即使全对，漏写有效范围也常丢分。

12. Summary and Revision Checklist | 总结与复习清单

By now you should be confident with: the formula for (a + b)ⁿ for positive integer n; calculating binomial coefficients using nCr or Pascal’s Triangle; finding a specific term using T_r+1; expanding rational powers using the series formula; stating and using the condition |x| < 1 for infinite expansions; and applying expansions to approximations and combined expressions.

到现在，你应当已经熟练掌握：正整数 n 时 (a + b)ⁿ 的公式；用 nCr 或帕斯卡三角形计算二项式系数；用 T_r+1 求特定项；用级数公式展开有理数次幂；陈述并使用无穷级数的 |x| < 1 条件；以及将展开式用于近似计算和组合表达式。

Review by attempting at least five past-paper questions, covering both SL and HL styles, and ensure you can write the validity range without prompting. This topic is a gateway to series expansions in calculus—master it now and strengthen your overall mathematical toolkit.

复习时请至少尝试五道往年真题，涵盖 SL 和 HL 风格，并确保你可以不经提示就写出有效范围。二项式展开是通向微积分中级数展开的大门——现在就拿下它，充实你的数学工具箱。

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IB WJEC Mathematics: Binomial Expansion – Key Points Revision | IB WJEC 数学：二项式展开 考点精讲