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

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

Binomial expansion is a core algebraic technique that expresses powers of a sum, (a+b)^n, as a polynomial. It appears in both IB and CIE A-level Mathematics, from finding coefficients in finite expansions to handling infinite series with restricted convergence.

二项式展开是将形如 (a+b)ⁿ 的幂展开为多项式的核心代数技巧。无论是 IB 还是 CIE A-level 数学，从有限展开中求系数，到处理具有收敛条件的无穷级数，二项式展开都占有重要地位。

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

A binomial is an algebraic expression containing two terms, such as (x + 3) or (2a – b). Expansion means writing the power of the binomial as a sum of terms of the form constant × a^? b^?.

二项式是包含两项的代数式，例如 (x + 3) 或 (2a – b) 。展开是指将二项式的幂写成形如常数 × a 的某次幂 × b 的某次幂的项之和。

When the exponent n is a positive integer, the expansion yields a finite polynomial with n+1 terms. When n is negative or a fraction, the expansion becomes an infinite series valid only for a certain range of x.

当指数 n 为正整数时，展开式给出一个包含 n+1 项的有限多项式。当 n 为负数或分数时，展开式变为一个仅在特定 x 范围内成立的无穷级数。

2. Pascal’s Triangle and Combination Formula | 帕斯卡三角形与组合公式

For positive integer exponents, coefficients can be read directly from Pascal’s triangle:

对于正整数指数，系数可以直接从帕斯卡三角形中读出：

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1
 1 5 10 10 5 1

Each number is the sum of the two above it, giving the coefficients of (a+b)^n for n = 0,1,2,… .

每个数都是其上方两数之和，给出了 (a+b)ⁿ 当 n = 0,1,2,… 时的系数。

Algebraically, the binomial coefficient is given by the combination formula C(n, r) = n! / [r!(n-r)!], where n! denotes n factorial. C(n, r) is read as “n choose r”.

代数上，二项式系数由组合公式 C(n, r) = n! / [r!(n-r)!] 给出，其中 n! 表示 n 的阶乘。C(n, r) 读作 “n 选 r”。

These coefficients are symmetric: C(n, r) = C(n, n-r).

这些系数具有对称性：C(n, r) = C(n, n-r)。

3. The General Term and Binomial Coefficients | 通项与二项式系数

For positive integer n, the expansion is (a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ.

对于正整数 n，展开式为 (a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ。

The general term, often denoted as the (r+1)th term, is T_{r+1} = C(n, r) aⁿ⁻ʳ bʳ, where r = 0,1,2,…,n.

通项通常指第 (r+1) 项，即 Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ，其中 r = 0,1,2,…,n。

Note the index: raising b to the power r and a to the power (n-r). This pattern is essential for solving term-finding problems.

注意指标：b 的指数为 r，a 的指数为 (n-r)。这一规律在解决求特定项问题中至关重要。

4. Expanding (a+b)^n for Positive Integer n | 正整数指数展开

To expand, determine a, b, and n, then apply the binomial theorem. For example, (2x – 3)⁴: here a = 2x, b = -3, n = 4.

展开时，先确定 a、b、n，再应用二项式定理。例如 (2x – 3)⁴ 中，a = 2x，b = -3，n = 4。

The expansion is: C(4,0)(2x)⁴ + C(4,1)(2x)³(-3) + C(4,2)(2x)²(-3)² + C(4,3)(2x)(-3)³ + C(4,4)(-3)⁴.

展开为：C(4,0)(2x)⁴ + C(4,1)(2x)³(-3) + C(4,2)(2x)²(-3)² + C(4,3)(2x)(-3)³ + C(4,4)(-3)⁴。

Simplify each term carefully. The coefficients are 1, 4, 6, 4, 1, giving 16x⁴ – 96x³ + 216x² – 216x + 81.

仔细化简每一项。系数为 1,4,6,4,1，结果为 16x⁴ – 96x³ + 216x² – 216x + 81。

Always pay attention to signs when b is negative.

当 b 为负数时，务必留意符号变化。

5. Finding Specific Terms and Coefficients | 求指定项与系数

Many exam questions ask for the coefficient of a particular power of x, such as “find the term in x⁵”. Use the general term formula T_{r+1} = C(n,r) aⁿ⁻ʳ bʳ, equate the power of x, and solve for r.

许多考题会要求求某特定 x 幂次的系数，例如“找出 x⁵ 的项”。使用通项公式 Tᵣ₊₁ = C(n,r) aⁿ⁻ʳ bʳ，令 x 的指数相等，解出 r。

For example, in (x² + 2/x)⁸, the term containing x⁴ is T_{r+1} = C(8,r) (x²)⁸⁻ʳ (2x⁻¹)ʳ = C(8,r) 2ʳ x¹⁶⁻²ʳ⁻ʳ = C(8,r) 2ʳ x¹⁶⁻³ʳ. Setting 16-3r = 4 gives r = 4. The coefficient is C(8,4) × 2⁴ = 70 × 16 = 1120.

例如，在 (x² + 2/x)⁸ 中，含 x⁴ 的项为 Tᵣ₊₁ = C(8,r) (x²)⁸⁻ʳ (2x⁻¹)ʳ = C(8,r) 2ʳ x¹⁶⁻²ʳ⁻ʳ = C(8,r) 2ʳ x¹⁶⁻³ʳ。令 16-3r = 4 得 r = 4。系数为 C(8,4) × 2⁴ = 70 × 16 = 1120。

If r is not an integer, such a term does not exist. This is a common trick question.

若 r 不是整数，则该类项不存在。这是常见的陷阱题。

6. Binomial Coefficients: Properties and Shortcuts | 二项式系数的性质与技巧

Key properties of C(n,r) include: symmetry C(n,r) = C(n,n-r); boundary values C(n,0) = C(n,n) = 1; Pascal’s rule C(n,r) + C(n,r+1) = C(n+1, r+1).

C(n,r) 的主要性质包括：对称性 C(n,r) = C(n,n-r)；边界值 C(n,0) = C(n,n) = 1；帕斯卡法则 C(n,r) + C(n,r+1) = C(n+1, r+1)。

The sum of all coefficients in the expansion of (a+b)ⁿ is obtained by setting a = b = 1, yielding 2ⁿ.

(a+b)ⁿ 展开式中各系数之和，可令 a = b = 1 得到，值为 2ⁿ。

Another useful trick: to find the sum of even-placed coefficients, use [(a+b)ⁿ + (a-b)ⁿ]/2 with a=1, b=1.

另一个有用技巧：求偶数项系数之和，可利用 [(a+b)ⁿ + (a-b)ⁿ]/2 并令 a=1, b=1。

7. Using Partial Fractions before Expansion | 先分部分分式再展开

When a rational expression looks like (px+q)/( (1+ax)(1+bx) ), it may be necessary to split it into partial fractions before applying the binomial expansion for negative/fractional powers.

当有理式形如 (px+q)/( (1+ax)(1+bx) ) 时，可能需要先将其拆为部分分式，再运用针对负指数或分数指数的二项式展开。

For instance, expand (3+x)/( (1-x)(1+2x) ) in ascending powers of x. First express as A/(1-x) + B/(1+2x), find A and B, then expand each term as (1-x)⁻¹ and (1+2x)⁻¹ valid for |x|<1 and |2x|<1 respectively.

例如，将 (3+x)/( (1-x)(1+2x) ) 按 x 的升幂展开。先表示为 A/(1-x) + B/(1+2x)，求出 A 和 B，然后将每一项分别展开为 (1-x)⁻¹ 和 (1+2x)⁻¹，它们分别当 |x|<1 和 |2x|<1 时成立。

The final expansion’s validity range is the intersection of the individual conditions, typically the stricter one.

最终展开式的有效范围是各条件之交集，通常取更严格的那个。

8. Binomial Expansion for Non-Positive Integer Powers | 非正整幂的二项展开

When n is not a positive integer (e.g., n = -2, 1/2, -3/4), the binomial expansion becomes an infinite series: (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + … .

当 n 不是正整数（如 n = -2, 1/2, -3/4）时，二项式展开变为无穷级数：(1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + … 。

For (a + bx)ⁿ, factor out a to get aⁿ (1 + (b/a)x)ⁿ, provided a ≠ 0 and |(b/a)x| < 1.

对于 (a + bx)ⁿ，需提取 a 得到 aⁿ (1 + (b/a)x)ⁿ，前提是 a ≠ 0 且 |(b/a)x| < 1。

The general term in this expansion is [n(n-1)…(n-r+1) / r!] xʳ. This resembles C(n,r) xʳ, where the binomial coefficient is defined for real n.

此展开式的通项为 [n(n-1)…(n-r+1) / r!] xʳ，类似于 C(n,r) xʳ，此时二项式系数对实数 n 有定义。

Do not rush through the sign; when n is negative, the factors alternate signs carefully.

勿要匆忙处理符号；当 n 为负数时，因子的符号需仔细交替。

9. Convergence Condition for Infinite Series | 无穷级数的收敛条件

The expansion of (1 + x)ⁿ as an infinite series converges only when |x| < 1. For (1 + kx)ⁿ, the condition becomes |kx| < 1, i.e., |x| < 1/|k|.

(1 + x)ⁿ 的无穷级数展开仅在 |x| < 1 时收敛。对于 (1 + kx)ⁿ，条件变为 |kx| < 1，即 |x| < 1/|k|。

When combining multiple expansions, the overall validity is the intersection of all such intervals. For example, if one term requires |x| < 1/2 and another requires |x| < 1/3, the series is valid for |x| < 1/3.

当组合多个展开式时，总体有效范围是所有区间的交集。例如，若一项要求 |x| < 1/2，另一项要求 |x| < 1/3，则该级数在 |x| < 1/3 时成立。

In exam questions, you are often asked to state the range of x for which the expansion is valid, so always check the modulus conditions.

考试中常要求说明展开式成立的 x 的取值范围，因此务必检查模条件。

10. Applications and Typical Exam Questions | 应用与典型考题

Common applications include: approximating square roots or cube roots using a binomial series (e.g., √(1+x) with x small), evaluating limits, and finding coefficients in products of series.

常见应用包括：利用二项级数逼近平方根或立方根（例如对很小的 x 展开 √(1+x)），求极限，以及确定两个级数乘积中的系数。

Example: Estimate √(4.1) by rewriting as 2(1 + 0.025)^(1/2) and using the first few terms of the expansion.

示例：将 √(4.1) 改写为 2(1 + 0.025)^(1/2)，然后利用展开式的前几项进行估值。

Another classic CIE/IB question: (a + bx)^c (1 + dx)^e. Multiply the relevant terms of both expansions to find the coefficient of a given power of x, paying attention to the ranges of x from each factor.

另一类经典 CIE/IB 题目：(a + bx)^c (1 + dx)^e。将两个展开式的相关项相乘，以找出某一 x 幂次的系数，并注意每个因式对应的 x 范围。

11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Forgetting the factorial in the denominator when generating coefficients for non-integer n is a frequent slip. Write out the n(n-1)/2! explicitly.

当 n 非整数时，生成系数时遗忘分母中的阶乘是常见疏忽。请明确写出 n(n-1)/2!。

Mixing up term index: in (a+b)ⁿ, the term containing bʳ corresponds to r, and the power of a is n-r. Always check which index the question requires.

混淆项指标：在 (a+b)ⁿ 中，含 bʳ 的项对应 r，而 a 的指数为 n-r。一定要检查问题要求哪一项指标。

Omitting the convergence interval: for infinite series, forgetting to state |x| < 1 (or < 1/|k|) loses marks. Underline the condition in your answer.

遗漏收敛区间：对于无穷级数，忘记写明 |x| < 1（或 < 1/|k|）会丢分。在答案中要强调该条件。

Sign errors when b is negative or when n is negative in the series. Use brackets systematically.

当 b 为负或级数中 n 为负时容易产生符号错误。系统地使用括号。

12. Summary and Exam Tips | 总结与考试技巧

Master the two main forms: finite expansion for positive integer n, and infinite series for non‑positive integer or rational n. The latter always requires a convergence check.

掌握两种主要形式：正整数 n 的有限展开，以及非正整数或有理数 n 的无穷级数展开。后者始终需要检查收敛条件。

When tackling any binomial question, first identify n, a, and b. Write down the general term immediately. For infinite cases, factor out constants to get the form (1 + u)ⁿ.

解答任何二项式题目时，首先确定 n、a 和 b。立刻写出通项。对于无穷情形，提取常数使其化为 (1 + u)ⁿ 的形式。

Practice finding individual coefficients without fully expanding the entire polynomial—this saves time and reduces errors.

练习直接求个别系数而不必完全展开整个多项式——这既省时又能减少错误。

Revise CIE P1, P2, and IB AA/AI HL binomial topics by re‑working past‑paper questions on term finding, validity ranges, and approximation problems. Consistent practice builds speed and accuracy.

复习 CIE P1、P2 以及 IB AA/AI HL 的二项式部分时，反复练习求项、有效范围和估值问题的真题。持续练习能提高速度和准确度。

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