📚 Binomial Expansion Key Points | 二项式展开考点精讲
In both IB and Edexcel mathematics, binomial expansion is a core algebraic skill that appears from basic integer powers through to infinite series with rational exponents. Mastering the notation, the general term, the validity conditions and common shortcuts can save valuable time and secure high marks on proof, coefficient-finding and approximation questions.
无论是 IB 还是 Edexcel 数学，二项式展开都是一项贯穿整数次幂到有理指数无穷级数的核心代数技能。熟练运用记号、通项公式、成立范围与常见技巧，不仅能节省大量答题时间，更能在证明、求系数和估值类题目中稳定拿分。
1. The Binomial Theorem for Non‑negative Integer n | 非负整数次幂的二项式定理
For any non‑ )ⁿ is a finite sum of (n + 1) terms. Each term contains a binomial coefficient C(n, r) = n! / [r! (n − r)!], a descending power of a and an ascending power of b.
对于任意非负整数 n 以及实数 a、b，(a + b)ⁿ 的展开式是一个含有 n+1 项的有限和。每一项包含二项式系数 C(n, r) = n! / [r! (n − r)!]、a 的降幂与 b 的升幂。
(a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ
You will also see the notation ⁿCᵣ or the bracket symbol (n r). In Edexcel and IB examinations all three notations are acceptable, but you must be consistent and clearly show factorial working when required.
考试中也可用 ⁿCᵣ 或 (n r) 记法。Edexcel 与 IB 对这三种记法都接受，但必须在需要时保持符号一致并清楚展示阶乘运算过程。
2. Pascal’s Triangle and Coefficient Patterns | 帕斯卡三角形与系数规律
iangle gives binomial coefficients row by row. Row n of the triangle lists C(n,0), C(n,1), … pattern helps with small n and reinforces the identity C(n, r) = C(n−1, r−1) + C(n−1, r).
帕斯卡三角形逐行给出二项式系数。第 n 行依次为 C(n,0), C(n,1), …, C(n,n)。这一直观模式对较小的 n 特别有用，同时也印证恒等式 C(n, r) = C(n−1, r−1) + C(n−1, r)。
can halve your work when finding coefficients or solving equations involving combinations.
考场上并不要求为大 n 画出整座三角形，但利用对称性 C(n, r) = C(n, n−r) 能使求系数或解组合方程的工作量减半。
3. The General Term Tᵣ₊₁ | 通项公式 Tᵣ₊₁
ing terms from r = 0 to r = n, the (r+1)th term in (a + b)ⁿ is given by Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ ecific term, such as the term independent of x, so the general term is the essential tool.
将项从 r = 0 编号至 r = n，(a + b)ⁿ 中第 (r+1) 项为 Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ。很多题目要求特定项（例如 x 的独立项），因此通项是解题的核心工具。
Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ
substitute into the general term, then collect powers of the variable of interest to find the required r.
当二项式包含单项式如 (2x + 3y²)ⁿ 时，设 a = 2x, b = 3y²，代入通项后整理目标变量的幂次，即可解出所需 r。
4. Symmetry and Useful Identities | 对称性与实用恒等式
as C(n, 5) = C(n, 9). Setting arguments equal or complementary yields n = 14 immediately.
对称性 C(n, r) = C(n, n−r) 不仅能加速展开，还能快速求解如 C(n, 5) = C(n, 9) 的方程。令下标相等或互补即得 n = 14。
The sum of all binomial coefficients in row n is 2ⁿ: C(n,0) + C(n,1) + … + C(n,n) = 2ⁿ.
第 n 行所有系数之和为 2ⁿ：C(n,0) + C(n,1) + … + C(n,n) = 2ⁿ。
The alternating sum gives 0: C(n,0) − C(n,1) + … + (−1)ⁿ C(n,n) = 0 (for n ≥ 1).
交错和为零：C(n,0) − C(n,1) + … + (−1)ⁿ C(n,n) = 0 (n ≥ 1)。
stions and can be verified by substituting x = 1 or x = −1 into the expansion of (1 + x)ⁿ.
这些恒等式经常出现在证明题中，只需将 x = 1 或 x = −1 代入 (1 + x)ⁿ 的展开式即可验证。
5. Extending to Negative and Fractional n | 推广至负指数与分数指数
When n is not a non‑negative integer, the binomial expansion becomes an infinite series. Writing the expression in the form (1 + x)ⁿ is crucial, and the expansion is valid only for |x| < 1.
当 n 不是非负整数时，二项式展开变为无穷级数。必须将表达式写成 (1 + x)ⁿ 的形式，且该展开仅在 |x| < 1 时成立。
(1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + … , |x| < 1
Here the coefficient of xʳ is n(n−1)…(n−r+1) / r!. Notice that the numerator has r factors, and factorials still appear in the denominator. The series does not terminate; you will only be asked for the first few terms.
此处 xʳ 的系数为 n(n−1)…(n−r+1) / r!。注意分子有 r 个因子，分母仍出现阶乘。级数不会自动终止，考试通常只要求写出前三至四项。
A common pitfall is forgetting to factor out a constant so that the bracket begins with 1. For example, (4 + 3x)⁻² must be rewritten as 4⁻² (1 + ¾ x)⁻² before expanding.
常见错误是忘记提取常数使括号内首项为 1。例如 (4 + 3x)⁻² 必须先改写成 4⁻² (1 + ¾ x)⁻² 后再展开。
6. Validity Interval and Convergence | 成立区间与收敛条件
For an expansion of the form (a + bx)ⁿ where n is fractional or negative, the validity condition is |bx/a| < 1, i.e. |x| < |a/b|. This condition must be stated alongside the series in any examination answer.
对于形如 (a + bx)ⁿ 的展开，当 n 为分数或负数时，成立条件为 |bx/a| < 1，即 |x| < |a/b|。任何考试答案都必须同时注明该不等式。
IB and Edexcel mark schemes specifically award marks for stating the range of x for which the expansion is valid. Even if the series is written perfectly, omitting |x| < … loses a mark.
IB 与 Edexcel 的评分标准明确规定，写出展开成立的 x 范围可得相应分值。即便级数书写完全正确，漏写 |x| < … 也会失分。
7. Approximations Using Binomial Expansions | 利用二项展开进行估值
Truncating a binomial series after two or three terms provides polynomial approximations of roots, reciprocals or powers. For instance, √1.01 ≈ 1 + ½(0.01) − ⅛(0.01)² yields 1.0049875.
截取二项级数的前两三项，即可得到根式、倒数或幂次的多项式近似。例如 √1.01 ≈ 1 + ½(0.01) − ⅛(0.01)² 给出 1.0049875。
√1 + x ≈ 1 + ½x − ⅛x², for |x| small
In exam questions you might be asked to approximate (something like ³√8.24) by rewriting as 2(1 + 0.03)⅓. Careful substitution and a quick check with a calculator will confirm the accuracy.
考试中可能要求估算类似于 ³√8.24 的值，通过改写为 2(1 + 0.03)⅓ 实现。仔细代换后可用计算器快速核对精度。
8. Finding Coefficients in Product Expansions | 乘积展开中的系数求解
When two binomials are multiplied, e.g. (1 + ax)(1 + bx)ⁿ, you typically expand (1 + bx)ⁿ up to the required power of x, then multiply by (1 + ax) and collect like terms. Identify the coefficient by equating the target term.
当两个二项式相乘时，例如 (1 + ax)(1 + bx)ⁿ，通常先展开 (1 + bx)ⁿ 至需要的 x 次幂，再乘以 (1 + ax) 合并同类项，通过对比目标项求出系数。
Questions may also involve dividing expansions or using partial fractions first. Always write the expansion in ascending powers of x and group coefficients systematically.
题目也可能涉及展开式相除或先分解为部分分式。务必将展开写成 x 的升幂形式并系统整理系数。
9. Typical Mistakes and How to Avoid Them | 常见错误与回避方法
Forgetting the validity condition – always check and state |x| < 1 (or equivalent) for non‑integer n.
忘记成立条件 – 非整数 n 时务必检查并注明 |x| < 1（或等价条件）。
Miswriting factorials for fractional n – use the product n(n−1)(n−2)…/r!; do not blindly write n!/(r!(n−r)!).
分数指数错误套用阶乘 – 应使用 n(n−1)(n−2)…/r!；不可直接套用 n!/(r!(n−r)!)。
Sign errors with negative n – write out the terms carefully: (1 + x)⁻¹ = 1 − x + x² − x³ …, alternating signs appear from the product of negative numbers.
负指数正负号错漏 – 仔细逐项展开：(1 + x)⁻¹ = 1 − x + x² − x³ …，负数连乘产生的交错正负号极易出错。
Not adjusting the bracket to (1 + …) form – factor out the constant first.
未调整为 (1 + …) 形式 – 必须先提取常数因子。
10. Worked Mini‑Examples | 精讲例题
Example 1: e coefficient of x³ in (2 + x)⁷. Set a = 2, b = x, n = 7. General term: Tᵣ₊₁ = C(7, r) 2⁷⁻ʳ xʳ. For x³, r = 3, coefficient = C(7, 3) × 2⁴ = 35 × 16 = 560.
例1：求 (2 + x)⁷ 中 x³ 的系数。令 a = 2, b = x, n = 7。通项 Tᵣ₊₁ = C(7, r) 2⁷⁻ʳ xʳ。x³ 对应 r = 3，系数 = C(7, 3) × 2⁴ = 35 × 16 = 560。
Example 2: ange of validity. Using (1 + X)ⁿ with X = −2x, n = −3: series = 1 + (−3)(−2x) + (−3)(−4)/2! (−2x)² + (−3)(−4)(−5)/3! (−2x)³ + … = 1 + 6x + 24x² + 80x³ + … Valid for |−2x| < 1 ⇒ |x| < ½.
例2：将 (1 − 2x)⁻³ 展开至 x³ 并注明成立范围。令 (1 + X)ⁿ，X = −2x, n = −3：展开 = 1 + 6x + 24x² + 80x³ + … 成立条件 |−2x| < 1 ⇒ |x| < ½。
Example 3: Use the expansion of (1 + x)½ to estimate √1.02. With x = 0.02, √1.02 ≈ 1 + ½(0.02) − ⅛(0.02)² = 1 + 0.01 − 0.00005 = 1.00995.
例3：利用 (1 + x)½ 的展开估算 √1.02。取 x = 0.02，√1.02 ≈ 1 + ½(0.02) − ⅛(0.02)² = 1.00995。
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