📚 GCSE Edexcel Mathematics: Binomial Expansion Key Points | GCSE Edexcel 数学：二项式展开 考点精讲
Binomial expansion is an essential part of the Edexcel GCSE Mathematics syllabus, often appearing in both non-calculator and calculator papers. Understanding how to expand expressions of the form (a + b)ⁿ where n is a positive integer is a key skill. This article provides a comprehensive revision guide, covering Pascal’s triangle, combination notation, specific term identification, and common exam pitfalls.
二项式展开是 Edexcel GCSE 数学大纲中不可或缺的一部分，常见于非计算器和计算器试卷中。理解如何展开形如 (a + b)ⁿ（n 为正整数）的表达式是一项关键技能。本文提供了一份全面的复习指南，涵盖帕斯卡三角形、组合数符号、特定项识别以及常见的考试陷阱。
1. What Is Binomial Expansion? | 什么是二项式展开？
A binomial is an algebraic expression containing two terms, such as (x + y) or (2a – 3b). Binomial expansion is the process of multiplying out a binomial raised to a positive integer power, expressing the result as a sum of terms involving powers of the two original terms. For example, (x + y)² = x² + 2xy + y².
二项式是一个包含两个项的代数表达式，例如 (x + y) 或 (2a – 3b)。二项式展开是将一个二项式进行正整数次幂的乘法运算，并将结果表示为包含原两项幂次的项的和的过程。例如，(x + y)² = x² + 2xy + y²。
2. Pascal’s Triangle: The Foundation | 帕斯卡三角形：基础
Pascal’s triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The rows correspond to the powers of a binomial expansion, starting with n = 0 at the top. The first few rows are: n = 0: 1; n = 1: 1 1; n = 2: 1 2 1; n = 3: 1 3 3 1; n = 4: 1 4 6 4 1. These numbers are the coefficients in the expansion of (a + b)ⁿ.
帕斯卡三角形是一个三角形的数字阵列，其中每个数字是其正上方两个数字之和。行对应于二项式展开的幂次，从顶部的 n = 0 开始。前几行是：n = 0: 1；n = 1: 1 1；n = 2: 1 2 1；n = 3: 1 3 3 1；n = 4: 1 4 6 4 1。这些数字正是 (a + b)ⁿ 展开式中的系数。
3. Using Pascal’s Triangle to Expand (a + b)ⁿ | 使用帕斯卡三角形展开 (a + b)ⁿ
To expand (a + b)ⁿ, first identify the nth row of Pascal’s triangle (remembering that the top row is n = 0). The coefficients are the entries in that row. Then, write down terms where the powers of a start at n and decrease by 1 each term, while the powers of b start at 0 and increase by 1 each term. For (x + 2)³, the row for n = 3 gives coefficients 1, 3, 3, 1. The expansion begins with 1·x³·2⁰ + 3·x²·2¹ + 3·x¹·2² + 1·x⁰·2³, which simplifies to x³ + 6x² + 12x + 8.
要展开 (a + b)ⁿ，首先找出帕斯卡三角形的第 n 行（记住顶行为 n = 0）。该行的数字即为系数。然后，写出各项，其中 a 的幂次从 n 开始，每项减少 1，而 b 的幂次从 0 开始，每项增加 1。对于 (x + 2)³，n = 3 的行给出系数 1, 3, 3, 1。展开式以 1·x³·2⁰ + 3·x²·2¹ + 3·x¹·2² + 1·x⁰·2³ 开始，化简得 x³ + 6x² + 12x + 8。
4. Introducing Combination Notation: ⁿCᵣ | 引入组合数符号：ⁿCᵣ
When n is large, drawing Pascal’s triangle becomes impractical. Instead, we use the combination formula or calculator function ⁿCᵣ (also written as nCr), which gives the coefficient of the term containing bʳ in the expansion of (a + b)ⁿ. The value of ⁿCᵣ is calculated as n! / [r!(n – r)!], but at GCSE level the calculator button is the main tool. For (a + b)⁵, the coefficients can be found using ⁵C₀ = 1, ⁵C₁ = 5, ⁵C₂ = 10, and so on.
当 n 较大时，绘制帕斯卡三角形变得不切实际。取而代之，我们使用组合数公式或计算器上的 ⁿCᵣ 函数（也写为 nCr），它给出 (a + b)ⁿ 展开式中含有 bʳ 的项的系数。ⁿCᵣ 的值计算为 n! / [r!(n – r)!]，但在 GCSE 阶段，计算器按钮是主要工具。对于 (a + b)⁵，系数可使用 ⁵C₀ = 1、⁵C₁ = 5、⁵C₂ = 10 等求得。
5. The General Term Formula | 通项公式
The expansion of (a + b)ⁿ can be written using sigma notation, but for exam questions it is more important to understand the general term: the (r + 1)th term is given by ⁿCᵣ · aⁿ⁻ʳ · bʳ, where r starts from 0. For example, the third term in the expansion of (x + 3)⁵ has r = 2: ⁵C₂ · x⁵⁻² · 3² = 10 · x³ · 9 = 90x³.
(a + b)ⁿ 的展开式可用求和符号 Σ 书写，但对考试来说更重要的是理解通项：第 (r + 1) 项为 ⁿCᵣ · aⁿ⁻ʳ · bʳ，其中 r 从 0 开始。例如，(x + 3)⁵ 展开式中的第三项，r = 2：⁵C₂ · x⁵⁻² · 3² = 10 · x³ · 9 = 90x³。
6. Expanding (1 + x)ⁿ | 展开 (1 + x)ⁿ
A common special case is (1 + x)ⁿ, where a = 1 and b = x. Since 1 raised to any power is 1, the expansion simplifies to: (1 + x)ⁿ = ⁿC₀ + ⁿC₁·x + ⁿC₂·x² + … + ⁿCₙ·xⁿ. This is often used to find coefficients of specific powers of x without writing the full expansion. For example, the coefficient of x³ in (1 + x)⁷ is ⁷C₃ = 35.
一个常见的特例是 (1 + x)ⁿ，其中 a = 1，b = x。由于 1 的任何次幂都是 1，展开式简化为：(1 + x)ⁿ = ⁿC₀ + ⁿC₁·x + ⁿC₂·x² + … + ⁿCₙ·xⁿ。这常被用来求特定 x 幂次的系数，而无需写出完整展开式。例如，(1 + x)⁷ 中 x³ 的系数为 ⁷C₃ = 35。
7. Expanding (a + bx)ⁿ | 展开 (a + bx)ⁿ
When the binomial contains a constant coefficient with x, such as (2 + 3x)⁴, treat 2 as ‘a’ and 3x as ‘b’. The general term is ⁿCᵣ · aⁿ⁻ʳ · (bx)ʳ. Remember to raise the entire ‘bx’ to the power r, including the constant. For (2 + 3x)⁴, the term in x² (r = 2) is ⁴C₂ · 2² · (3x)² = 6 · 4 · 9x² = 216x².
当二项式中 x 带有常系数时，例如 (2 + 3x)⁴，将 2 视为 ‘a’，3x 视为 ‘b’。通项为 ⁿCᵣ · aⁿ⁻ʳ · (bx)ʳ。切记将整个 ‘bx’ 乘方 r 次，包括常数系数。对于 (2 + 3x)⁴，含 x² 的项（r = 2）为 ⁴C₂ · 2² · (3x)² = 6 · 4 · 9x² = 216x²。
8. Dealing with Negative Terms | 负项的处理
If the binomial is a subtraction, such as (x – 4)⁵, rewrite it as (x + (-4))⁵. Then apply the general term with b = -4. The signs of the terms will alternate because (-4)ʳ is positive when r is even and negative when r is odd. For (2x – 3)⁴, the expansion is 1·(2x)⁴·(-3)⁰ + 4·(2x)³·(-3)¹ + 6·(2x)²·(-3)² + 4·(2x)¹·(-3)³ + 1·(2x)⁰·(-3)⁴, which simplifies to 16x⁴ – 96x³ + 216x² – 216x + 81.
如果二项式是减法，例如 (x – 4)⁵，将其改写为 (x + (-4))⁵。然后应用通项，令 b = -4。各项的符号会交替出现，因为 (-4)ʳ 当 r 为偶数时为正，当 r 为奇数时为负。对于 (2x – 3)⁴，展开式为 1·(2x)⁴·(-3)⁰ + 4·(2x)³·(-3)¹ + 6·(2x)²·(-3)² + 4·(2x)¹·(-3)³ + 1·(2x)⁰·(-3)⁴，化简得 16x⁴ – 96x³ + 216x² – 216x + 81。
9. Finding a Specific Coefficient or Term | 求特定系数或项
Exam questions frequently ask for the coefficient of a particular power of x without requiring the full expansion. Use the general term formula, set the power of x equal to the required power, solve for r, and then compute the coefficient. For instance, to find the coefficient of x⁵ in (2 + x)⁸: the term with x⁵ occurs when r = 5. The term is ⁸C₅ · 2³ · x⁵ = 56 · 8 · x⁵ = 448x⁵, so the coefficient is 448.
考试题目经常要求求特定 x 幂次的系数，而不要求完整展开。使用通项公式，令 x 的幂次等于所需幂次，解出 r，然后计算系数。例如，求 (2 + x)⁸ 中 x⁵ 的系数：含 x⁵ 的项出现在 r = 5。该项为 ⁸C₅ · 2³ · x⁵ = 56 · 8 · x⁵ = 448x⁵，因此系数为 448。
10. Calculator Skills for Binomial Expansion | 二项式展开的计算器技能
On a GCSE Edexcel calculator, the nCr function is usually accessed via the probability menu (often labeled ‘nCr’ or ‘C’). Check your calculator model, but the typical steps are: type n, press the nCr button (or shift + a button), then type r, and press equals. For example, to compute ¹⁰C₃, input 10 nCr 3, which gives 120. Always double-check that you have entered n before r.
在 GCSE Edexcel 计算器上，nCr 功能通常通过概率菜单访问（通常标记为 ‘nCr’ 或 ‘C’）。请查看您的计算器型号，但典型步骤是：输入 n，按下 nCr 按钮（或 shift + 某个按钮），然后输入 r，再按等号。例如，要计算 ¹⁰C₃，输入 10 nCr 3，得到 120。请务必检查您是否先输入了 n 再输入 r。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One frequent error is applying powers incorrectly to the constant within (bx). For (3x)², students sometimes write 3x² instead of 9x². Another mistake is forgetting that the binomial expansion has (n + 1) terms, not n terms. Also, when using the general term, mixing up the roles of a and b leads to wrong coefficients. Always write down the term template with ⁿCᵣ · aⁿ⁻ʳ · bʳ before substituting values.
一个常见错误是将 (bx) 中的常数进行幂运算时出错。对于 (3x)²，学生有时会写成 3x² 而不是 9x²。另一个错误是忘记二项式展开有 (n + 1) 项，而非 n 项。此外，在使用通项时，混淆 a 和 b 的角色会导致系数错误。在代入数值之前，务必先写下带有 ⁿCᵣ · aⁿ⁻ʳ · bʳ 的项模板。
12. Summary and Revision Tips | 总结与复习提示
Master binomial expansion by practising with both Pascal’s triangle (for small n) and the nCr method (for any n). Memorise the general term formula, and become fluent with your calculator’s nCr function. Work through past paper questions that ask for specific coefficients, especially those involving negative and fractional constants inside the bracket. With consistent practice, binomial expansion questions become reliable marks on your GCSE exam.
通过使用帕斯卡三角形（适用于较小的 n）和 nCr 方法（适用于所有 n）进行练习，从而掌握二项式展开。牢记通项公式，并熟练使用计算器的 nCr 功能。练习往年的真题，尤其是那些要求特定系数、且括号内含有负常数或分数常数的题目。通过持续练习，二项式展开题将成为你 GCSE 考试中稳定的得分点。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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