IB Maths: Binomial Expansion – Key Exam Points | IB 数学：二项式展开考点精讲

📚 IB Maths: Binomial Expansion – Key Exam Points | IB 数学：二项式展开考点精讲

The binomial expansion is a cornerstone of IB Mathematics, bridging algebra, combinatorics, and probability. Mastering it requires not only memorising the theorem but also understanding its underlying structure, from Pascal’s triangle to the general term formula. This revision guide covers essential exam techniques, common pitfalls, and advanced applications—including rational exponents for HL students—to help you tackle any binomial question with confidence.

二项式展开是 IB 数学的基石，连接代数、组合数学与概率。要掌握它，不仅需要熟记定理，还要理解其底层结构——从帕斯卡三角形到一般项公式。本文精讲涵盖重要的考试技巧、常见错误以及进阶应用（包括 HL 要求的有理指数展开），帮助你自信应对任何二项式题目。

1. Introduction to the Binomial Theorem | 二项式定理简介

The binomial theorem states that for any positive integer n, the expansion of (a + b)ⁿ is the sum of terms of the form C(n, r) a^n−r b^r, where r runs from 0 to n. This compact formula forms the basis for all standard expansions tested in IB exams.

二项式定理指出，对于任意正整数 n，(a + b)ⁿ 的展开式是各项 C(n, r) a^n−r b^r 之和，其中 r 从 0 取到 n。这个简洁的公式是 IB 考试中所有标准展开问题的基础。

The coefficients C(n, r) are called binomial coefficients, and they can be read directly from Pascal’s triangle or calculated using factorials. The full expansion can be written as:

系数 C(n, r) 称为二项式系数，可以直接从帕斯卡三角形读出，或用阶乘计算。完整展开式可写成：

(a + b)ⁿ = Σ_r=0ⁿ C(n, r) a^n−r b^r

IB questions often ask for the expansion up to a specific term, or to find a term containing a certain power of x. Understanding the sigma notation and the index r is the first step to solving these problems systematically.

IB 考题经常要求展开到特定项，或者找出包含 x 的某次幂的项。理解求和符号 Σ 和指标 r，是系统解决此类问题的第一步。

2. Pascal’s Triangle and Combinatorial Numbers | 帕斯卡三角形与组合数

Pascal’s triangle gives a visual pattern of the binomial coefficients for small values of n. Each entry is the sum of the two entries directly above it, which mirrors the identity C(n, r) = C(n−1, r−1) + C(n−1, r).

帕斯卡三角形为较小的 n 提供了二项式系数的直观模式。每个数等于它肩上两个数之和，这正好反映了恒等式 C(n, r) = C(n−1, r−1) + C(n−1, r)。

IB students must be able to relate Pascal’s triangle rows to the expansion of (a + b)ⁿ. For instance, the row 1, 5, 10, 10, 5, 1 corresponds to n = 5. This relationship often appears in exam questions involving compound events or simple coefficient checks.

IB 学生必须能将帕斯卡三角形的行与 (a + b)ⁿ 的展开联系起来。例如，行 1, 5, 10, 10, 5, 1 对应 n = 5。这种关系经常出现在涉及复合事件或简单系数验算的考试题中。

The binomial coefficient C(n, r) is also read as “n choose r” and is given by the formula:

二项式系数 C(n, r) 也读作“n 选 r”，其计算公式为：

C(n, r) = n! / (r! × (n−r)!)

On both SL and HL papers, you must be comfortable evaluating C(n, r) using a calculator’s nCr function, or manually for small numbers. Many candidates lose marks by mixing up r and n−r in the exponent.

在 SL 与 HL 试卷中，你必须熟练使用计算器上的 nCr 功能求值，或小数字时手算。很多考生因为在指数中混淆 r 和 n−r 而丢分。

3. The General Term Formula | 一般项公式

The most powerful exam tool for binomial expansion is the general term formula. The (r + 1)^th term in the expansion of (a + b)ⁿ is T_r+1 = C(n, r) a^n−r b^r. Note that the index r starts at 0 for the first term.

二项式展开最有用的考试工具是一般项公式。(a + b)ⁿ 展开式的第 (r + 1) 项为 T_r+1 = C(n, r) a^n−r b^r。注意第一个项的指标 r 从 0 开始。

If the given expression is something like (2x³ − 1/x)¹⁰, treat a = 2x³ and b = −1/x. Then the general term becomes C(10, r) (2x³)^10−r (−1/x)^r. Simplify the powers of x carefully to form the exponent equation needed to find a specific term.

如果给出的式子是类似 (2x³ − 1/x)¹⁰，将 a 视为 2x³，b 视为 −1/x。那么一般项就变成 C(10, r) (2x³)^10−r (−1/x)^r。仔细化简 x 的幂次，列出指数方程，就能定位到所求的特定项。

Always write out T_r+1 explicitly before solving for r. This approach prevents sign errors and ensures you capture coefficients correctly, especially when b contains a minus sign or a coefficient like −3.

在解出 r 之前，一定要先用文字写出 T_r+1。这样做可以避免符号错误，并确保正确提取系数，尤其当 b 含有负号或系数如 −3 时。

4. Finding Specific Coefficients | 求特定系数

A typical IB question asks: “Find the coefficient of x^k in the expansion of (px + qy)ⁿ.” To answer this, set up the general term, simplify the power of x to k, solve for r, and then evaluate the coefficient including all constants and signs.

典型的 IB 考题是：“求 (px + qy)ⁿ 的展开式中 x^k 的系数”。解法是：建立一般项，化简 x 的幂次等于 k，解出 r，然后求值系数，包括所有常数和符号。

Example: Find the coefficient of x⁴ in (2x − 3/x²)⁸. Here the term is C(8, r) (2x)^8−r (−3/x²)^r. Simplifying x exponent: (8−r) − 2r = 8 − 3r. Set 8 − 3r = 4, giving r = 4/3, which is not an integer. Therefore, the coefficient of x⁴ is 0 because that term does not exist. IB exams love testing this logical conclusion.

举例：求 (2x − 3/x²)⁸ 展开式中 x⁴ 的系数。此时项为 C(8, r) (2x)^8−r (−3/x²)^r。化简 x 指数：(8−r) − 2r = 8 − 3r。设 8 − 3r = 4，得 r = 4/3，不是整数。因此，x⁴ 的系数为 0，因为该项不存在。IB 考试喜欢考查这种逻辑结论。

When r is an integer, substitute back to calculate the full coefficient: C(8, r) × 2^8−r × (−3)^r. Pay careful attention to alternating signs—a negative base raised to an odd power remains negative, often causing sign errors under exam pressure.

当 r 为整数时，代回计算完整系数：C(8, r) × 2^8−r × (−3)^r。仔细注意交替符号——底数为负数的奇次幂仍为负，这在考试压力下常引发符号错误。

5. Finding the Constant Term | 求常数项

The constant term appears when the net power of x is zero. Set the simplified exponent of x in the general term equal to 0 and solve for r. This is one of the most frequently examined skills in IB HL and SL Papers 1 and 2.

常数项出现在 x 的净指数为零时。将一般项中 x 的化简指数设为零，解出 r。这是 IB HL 和 SL 试卷 1、2 中最常考查的技能之一。

For example, to find the constant term in (x² + 2/x)⁹: T_r+1 = C(9, r) (x²)^9−r (2/x)^r. The power of x is 2(9−r) − r = 18 − 3r. Setting 18 − 3r = 0 yields r = 6. Then the constant term is C(9, 6) × 2⁶ = 84 × 64 = 5376. Notice that the coefficient includes the 2^r factor—a detail many overlook.

例如，求 (x² + 2/x)⁹ 的常数项：T_r+1 = C(9, r) (x²)^9−r (2/x)^r。x 的指数为 2(9−r) − r = 18 − 3r。令 18 − 3r = 0，得 r = 6。常数项为 C(9, 6) × 2⁶ = 84 × 64 = 5376。注意系数包含 2^r 因子——这是许多人忽略的细节。

The trick is always to isolate the x-power, solve for r, and only then evaluate the full term. If r is not an integer or lies outside the range 0 to n, state that there is no constant term. IB mark schemes reward this reasoning explicitly.

诀窍是先把 x 的幂单独分离出来，解出 r，然后才计算整个项的值。如果 r 不是整数，或超出 0 到 n 的范围，则说明不存在常数项。IB 评分标准明确奖励这种推理。

6. Sum of Coefficients and Symmetry Properties | 系数和与对称性

The sum of all coefficients in the expansion of (a + bx)ⁿ is obtained by setting x = 1 and any other variables to 1. For example, the sum of coefficients in (3 + 2x)⁵ is (3 + 2)⁵ = 5⁵ = 3125. This quick trick appears on multiple-choice and short-answer sections.

展开式 (a + bx)ⁿ 中所有系数之和，可通过令 x = 1 并将其他变量设为 1 求得。例如，(3 + 2x)⁵ 的系数和为 (3 + 2)⁵ = 5⁵ = 3125。这个快捷技巧常用于选择题和简答题。

Symmetry property: C(n, r) = C(n, n−r). This means the coefficients read the same forwards and backwards. In an expansion like (1 + x)⁶, the third term from the start and the third term from the end share the same coefficient. This can serve as a consistency check in exams.

对称性：C(n, r) = C(n, n−r)。这意味着系数正反向相同。在 (1 + x)⁶ 的展开式中，从开始数第三项和从末尾数第三项系数相同。这可以作为考试中检查一致性的方法。

Another useful property is the sum of coefficients of odd powers versus even powers. By evaluating the binomial expression at x = 1 and at x = −1, you can find the sum of coefficients of the even-powered terms and odd-powered terms separately. This is an IB HL favourite.

另一个有用性质是奇次幂系数之和与偶次幂系数之和。通过在 x = 1 和 x = −1 处求二项式表达式的值，可以分别求出偶次幂项和奇次幂项的系数之和。这是 IB HL 特别爱考的内容。

Sum of all coefficients: f(1) | Sum of even terms: [f(1) + f(−1)] / 2

7. Binomial Expansion of Three Terms | 三项式的展开技巧

Questions like “Find the coefficient of x² in (1 + x + x²)⁵” treat the expression as a trinomial. The most systematic IB method is to rewrite it as [(1 + x) + x²]⁵ and apply the binomial theorem twice, or to use the multinomial expansion. However, for low powers, choosing pairs of terms that multiply to give x² is more efficient.

像“求 (1 + x + x²)⁵ 中 x² 的系数”这类题目，将式子视为三项式。IB 中最系统的方法是将其改写为 [(1 + x) + x²]⁵ 并两次应用二项式定理，或者使用多项式展开。但对于低次幂，挑选能乘出 x² 的项对更高效率。

Using the binomial theorem approach: expand as Σ C(5, k) (1 + x)^5−k (x²)^k. Then expand (1 + x)^5−k and collect terms where the total power of x is 2. This structured method prevents missing contributions from different branches of the expansion.

使用二项式定理方法：展开为 Σ C(5, k) (1 + x)^5−k (x²)^k。再展开 (1 + x)^5−k，收集 x 的总次数为 2 的项。这种结构化方法可避免遗漏展开不同分支的贡献。

IB exams often limit trinomial questions to relatively small n, where listing combinations is manageable. For instance, to get x², you might take two factors of x from the x-term and none from x², or one factor from x² and nothing else. Practice these combinational approaches to save time.

IB 考试通常将三项式题目限制在较小的 n，使得列出组合可行。例如，要得到 x²，你可以从 x 项取两个因子而不取 x²，或者从 x² 取一个因子，其余不动。练习这些组合方法以节省时间。

8. Relationship with Binomial Distribution | 与二项分布的关系

The binomial expansion and the binomial probability distribution share the same coefficients C(n, r). In the binomial distribution B(n, p), the probability of exactly r successes is C(n, r) p^r (1−p)^n−r. This is mathematically identical to the (r+1)th term of (q + p)ⁿ where q = 1−p.

二项式展开与二项概率分布共享相同的系数 C(n, r)。在二项分布 B(n, p) 中，恰好 r 次成功的概率为 C(n, r) p^r (1−p)^n−r。这在数学上与 (q + p)ⁿ 的第 (r+1) 项相同，其中 q = 1−p。

IB questions may link the two concepts, for instance by asking for the sum of certain binomial coefficients and interpreting it as a probability. Recognising that Σ C(n, r) = 2ⁿ corresponds to the total number of subsets of an n-element set, which also equals the sum of all probabilities when p = 1/2.

IB 题目可能将这两个概念联系起来，例如求某些二项式系数的和，并将其解释为概率。意识到 Σ C(n, r) = 2ⁿ 对应 n 元集合的子集总数，同时也等于 p = 1/2 时所有概率之和。

Occasionally, exam questions ask: “Use the binomial theorem to prove that the sum of binomial probabilities is 1.” This is a straightforward application: [p + (1−p)]ⁿ = 1ⁿ = 1. Understanding this identity can earn easy marks on Paper 2.

有时考题要求：“利用二项式定理证明二项概率之和为 1”。这是个直接的应用：[p + (1−p)]ⁿ = 1ⁿ = 1。理解这个恒等式可以在试卷 2 中轻松得分。

9. Binomial Series for Rational Exponents (HL extension) | 有理指数二项式级数（HL 拓展）

For IB Mathematics: Analysis and Approaches HL, the binomial expansion extends to rational exponents n, where |x| < 1 for convergence. The formula (1 + x)ⁿ = 1 + n x + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … is an infinite series.

对于 IB 数学：分析与方法 HL，二项式展开扩展到有理指数 n，此时需 |x| < 1 保证收敛。公式 (1 + x)ⁿ = 1 + n x + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … 是一个无穷级数。

HL students must state the validity condition |x| < 1 (or < 1 for the modulus) and recognise that the expansion is an approximation unless the series terminates. Questions may ask for the first four terms in the expansion of (1 − 2x)⁻¹ or √(4 + x) rewritten as 2(1 + x/4)^1/2.

HL 学生必须写明有效性条件 |x| < 1（或模长 < 1），并认识到除非级数终止，否则展开式是近似值。考题可能要求写出 (1 − 2x)⁻¹ 或 √(4 + x) 重写为 2(1 + x/4)^1/2 的前四项。

When n is a fraction or negative integer, the coefficient formula uses the generalised binomial coefficient: C(n, r) = n(n−1)(n−2)…(n−r+1) / r!. Be careful not to use the nCr button for non-integer n—it doesn’t apply. HL mark schemes explicitly test the step where the range of validity is stated or used to justify discarding higher terms.

当 n 为分数或负整数时，系数公式使用推广的二项式系数：C(n, r) = n(n−1)(n−2)…(n−r+1) / r!。注意不要对非整数 n 使用 nCr 按钮——这不适用。HL 评分标准明确测试写明有效性范围或用以证明可舍弃高阶项的步骤。

Expansions like (a + bx)ⁿ must first be factored into aⁿ (1 + (b/a) x)ⁿ to match the standard form. The range then becomes |(b/a) x| < 1, i.e. |x| < |a/b|. Missing this manipulation leads to an invalid expansion set and lost marks.

像 (a + bx)ⁿ 这样的展开式，必须先因式分解为 aⁿ (1 + (b/a) x)ⁿ，以符合标准形式。有效性范围随即变为 |(b/a) x| < 1，即 |x| < |a/b|。遗漏这一步会导致展开设定无效而失分。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Mistake 1: Forgetting that the index r corresponds to the (r+1)th term, not the rth term. Always write T_r+1 = C(n, r) a^n−r b^r and double-check your term number in the final answer.

错误一：忘记指标 r 对应第 (r+1) 项，而非第 r 项。务必写出 T_r+1 = C(n, r) a^n−r b^r，并在最终答案中复查项序号。

Mistake 2: Losing the coefficient from the a or b term. If a = 2x, then a^n−r = (2x)^n−r = 2^n−r x^n−r. Both the constant and the variable must be raised to the power. Write brackets around the whole term before applying the exponent.

错误二：遗漏 a 或 b 项中的系数。若 a = 2x，则 a^n−r = (2x)^n−r = 2^n−r x^n−r。常量和变量都要升幂。在使用指数前，先将整个项加上括号。

Mistake 3: Sign errors with negative b. If b = −3y, then b^r = (−3y)^r = (−1)^r 3^r y^r. The sign alternates with r. Use a table or add an extra check step for the sign of each term when writing out the full expansion.

错误三：b 为负时的符号错误。若 b = −3y，则 b^r = (−3y)^r = (−1)^r 3^r y^r。符号随 r 交替变化。在写出完整展开式时，用表格或添加额外检查步骤确认每项的符号。

Mistake 4: For HL, applying the nCr formula for non-integer n. Use the factorial-like general coefficient formula, and always state the validity range. Without it, you might lose method marks even if the expansion is correct.

错误四：HL 中对非整数 n 使用 nCr 公式。应使用类似阶乘的推广系数公式，并始终写明有效范围。否则即使展开正确，也可能丢掉方法分。

Final tip: Practise past paper questions under timed conditions. Typical questions: “Find the term independent of x”, “Show that the coefficient of x³ is…”, or “Hence evaluate the sum…”. Structure your working clearly, label the general term, and always box your final answer for visibility.

最后建议：限时练习历年真题。常见题型：“求与 x 无关的项”“证明 x³ 的系数为……”“由此求值……”。清晰地写出做题步骤，标记一般项，并始终框出最终答案以便阅卷。

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IB Maths: Binomial Expansion – Key Exam Points | IB 数学：二项式展开 考点精讲