A-Level Maths: Expansion of Brackets in Algebra | A-Level 数学：括号展开的代数运算

📚 A-Level Maths: Expansion of Brackets in Algebra | A-Level 数学：括号展开的代数运算

Expanding brackets is one of the most fundamental algebraic skills tested at A-Level. It appears everywhere: solving equations, sketching graphs, differentiating polynomials, and proving identities. Without confident bracket expansion, later topics such as factorising, functions, logarithms, and calculus become unnecessarily difficult.

括号展开是 A-Level 数学中最基础、最重要的代数技能之一。它几乎贯穿所有章节：解方程、画函数图像、多项式求导、证明恒等式等。如果括号展开不熟练，后续学习因式分解、函数、对数以及微积分时都会遇到不必要的困难。

Mastering this skill is not about memorising tricks; it is about understanding the distributive law and applying it systematically. This article covers single brackets, double brackets, perfect squares, difference of two squares, triple brackets, Pascal’s triangle, and the common pitfalls students face.

掌握括号展开不是靠死记硬背技巧，而是理解分配律并系统化地应用。这篇文章将覆盖单括号、双括号、完全平方、平方差、三重括号、帕斯卡三角以及学生常见的错误。

1. Why Expanding Brackets Matters | 为什么展开括号很重要

In algebra, a bracket groups terms together. Expanding means removing the brackets by multiplying every term inside the bracket by the term or expression outside. The result is a sum of terms with no brackets remaining.

在代数中，括号把若干项组合在一起。展开的意思是通过将括号外面的项逐一乘以括号里的每一项，从而去掉括号。最终结果是一个不含括号的多项式之和。

For example, the expression 3(x + 2) means “3 lots of (x + 2)”, which is equal to 3x + 6. Expansion transforms a compact product into an extended sum, which is often easier to manipulate when solving equations or simplifying expressions.

例如，3(x + 2) 表示“3 倍的 (x + 2)”，等于 3x + 6。展开将一个紧凑的乘积变成一个扩展后的和，这在解方程或化简表达式时通常更容易操作。

Expansion is also the reverse of factorisation. If you understand how to expand confidently, you will find factorisation more intuitive, because you can quickly check whether a factorised answer is correct by expanding it again.

展开也是因式分解的逆过程。如果你能熟练展开，因式分解就会变得更直观，因为你可以通过再次展开来快速检验因式分解的结果是否正确。

2. Removing a Single Bracket | 去括号：单项式乘多项式

The basic rule for a single bracket is the distributive law: for any numbers a, b and c,

单括号展开的基本规则是分配律：对任意数 a、b、c，有

a(b + c) = ab + ac

This means the term outside the bracket must be multiplied by every term inside the bracket. It is a common mistake to multiply only the first term inside the bracket.

这意味着括号外面的项必须乘以括号里面每一个项。常见错误是只乘括号里的第一项。

Example: Expand 4(3x – 7).

示例：展开 4(3x – 7)。

4(3x – 7) = 4 × 3x + 4 × (-7) = 12x – 28

When the term outside is negative, pay careful attention to signs. For instance,

当外面的项是负数时，要特别注意符号。例如，

-2x(3x² – 4x + 1) = -6x³ + 8x² – 2x

Here -2x multiplies 3x² to give -6x³, then multiplies -4x to give +8x², and finally multiplies +1 to give -2x. Every sign is determined by the usual rules of integers.

这里 -2x 乘以 3x² 得到 -6x³，乘以 -4x 得到 +8x²，最后乘以 +1 得到 -2x。每个符号都由整数乘法的符号法则决定。

3. Multiplying Two Linear Brackets | 二次展开：两个一次括号相乘

To expand two brackets such as (A + B)(C + D), every term in the first bracket must multiply every term in the second bracket. The general pattern is:

要展开两个括号，例如 (A + B)(C + D)，必须将第一个括号中的每一项与第二个括号中的每一项相乘。一般模式是：

(A + B)(C + D) = AC + AD + BC + BD

Example: Expand (x + 4)(x – 3).

示例：展开 (x + 4)(x – 3)。

(x + 4)(x – 3) = x² – 3x + 4x – 12 = x² + x – 12

Notice that -3x and +4x are like terms, so they are combined to give +x. After expanding two linear brackets, the result is usually a quadratic expression in the form ax² + bx + c.

注意 -3x 与 +4x 是同类项，合并后得到 +x。展开两个一次括号后，结果通常是一个形如 ax² + bx + c 的二次表达式。

4. The FOIL Method in Detail | FOIL 方法详解

For two binomials, the FOIL method organises the four multiplications. FOIL stands for First, Outer, Inner, Last.

对于两个二项式相乘，FOIL 方法将四次乘法整理清楚。FOIL 分别代表 First（第一项）、Outer（外项）、Inner（内项）、Last（最后一项）。

First: Multiply the first terms of each bracket.

第一项：将两个括号的第一项相乘。
Outer: Multiply the outer terms of the product.

外项：将乘积中的外侧两项相乘。
Inner: Multiply the inner terms of the product.

内项：将乘积中的内侧两项相乘。
Last: Multiply the last terms of each bracket.

最后一项：将两个括号的最后一项相乘。

Example: Expand (2x + 3)(x + 5).

示例：展开 (2x + 3)(x + 5)。

First: 2x × x = 2x².

第一项：2x × x = 2x²。
Outer: 2x × 5 = 10x.

外项：2x × 5 = 10x。
Inner: 3 × x = 3x.

内项：3 × x = 3x。
Last: 3 × 5 = 15.

最后一项：3 × 5 = 15。

Combining the results gives:

合并结果得到：

(2x + 3)(x + 5) = 2x² + 10x + 3x + 15 = 2x² + 13x + 15

The Outer and Inner products are often like terms, so they must be added together in the final expression.

外项乘积和内项乘积通常是同类项，因此最终表达式中必须将它们相加合并。

5. Perfect Squares | 完全平方公式

A perfect square is a binomial multiplied by itself. The two standard identities are:

完全平方是一个二项式与自身相乘。两个标准公式是：

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab + b²

Example: Expand (3x – 2)².

示例：展开 (3x – 2)²。

Using the second identity with a = 3x and b = 2:

利用第二个公式，令 a = 3x，b = 2：

(3x – 2)² = (3x)² – 2(3x)(2) + 2² = 9x² – 12x + 4

A very common error is to write (x + 3)² = x² + 9. This is wrong because the middle term 2ab is missing. Always write out the squared binomial as a product if you are unsure, or remember the middle term.

一个非常常见的错误是写成 (x + 3)² = x² + 9。这是错误的，因为缺少中间项 2ab。如果不太确定，可以先把平方写成两个相同括号相乘；否则一定要记得中间项。

6. Difference of Two Squares | 平方差公式

When two brackets are identical except for the sign between the terms, their expansion simplifies dramatically:

当两个括号除了中间正负号不同之外完全相同时，它们的展开会大大简化：

(a + b)(a – b) = a² – b²

Example 1: Expand (x + 5)(x – 5).

示例 1：展开 (x + 5)(x – 5)。

(x + 5)(x – 5) = x² – 25

The middle terms +5x and -5x cancel each other out, so only two terms remain.

中间项 +5x 与 -5x 相互抵消，因此只剩两项。

Example 2: Expand (2x + 3)(2x – 3).

示例 2：展开 (2x + 3)(2x – 3)。

(2x + 3)(2x – 3) = (2x)² – 3² = 4x² – 9

Notice that the square applies to the whole term 2x, giving 4x², not 2x². This identity is also extremely useful for factorising expressions such as x² – 16 and for simplifying algebraic fractions.

注意平方作用于整个项 2x，得到 4x²，而不是 2x²。这个公式在因式分解 x² – 16 以及化简代数分式时也非常有用。

7. Expanding Three or More Brackets | 三个及以上括号的展开

To expand three brackets, begin by multiplying two of them, then multiply the resulting expression by the third bracket. The order can be chosen to make the calculation easiest, but the simplest method is usually to expand the first two brackets first.

展开三个括号时，先把其中两个相乘，再把得到的表达式与第三个括号相乘。计算顺序可以自由选择以方便为准，但通常最简单的做法是先展开前两个括号。

Example: Expand (x + 1)(x + 2)(x + 3).

示例：展开 (x + 1)(x + 2)(x + 3)。

First multiply the first two brackets:

先将前两个括号相乘：

(x + 1)(x + 2) = x² + 3x + 2

Then multiply by the third bracket:

再与第三个括号相乘：

(x² + 3x + 2)(x + 3) = x³ + 6x² + 11x + 6

Check the expansion step by step: x² multiplied by x gives x³; x² multiplied by 3 gives 3x²; 3x multiplied by x gives 3x²; 3x multiplied by 3 gives 9x; 2 multiplied by x gives 2x; 2 multiplied by 3 gives 6. Collecting like terms gives x³ + 6x² + 11x + 6.

逐步检查展开过程：x² 乘以 x 得 x³；x² 乘以 3 得 3x²；3x 乘以 x 得 3x²；3x 乘以 3 得 9x；2 乘以 x 得 2x；2 乘以 3 得 6。合并同类项得到 x³ + 6x² + 11x + 6。

For expressions with four or more brackets, use the same step-by-step approach. Expanding systematically is always safer than trying to multiply everything at once.

对于四个或更多括号的表达式，仍然使用同样的逐步方法。系统化逐步展开永远比试图一次性乘完更安全。

8. Using Pascal’s Triangle | 使用帕斯卡三角

When expanding a binomial raised to a positive integer power, Pascal’s triangle provides the coefficients directly. The rows of Pascal’s triangle start as follows:

当展开一个二项式正整数次幂时，帕斯卡三角可以直接提供各项系数。帕斯卡三角的开头几行如下：

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 coefficients apply when expanding (a + b)ⁿ. For example, using row n = 3:

这些系数用于展开 (a + b)ⁿ。例如，使用 n = 3 这一行：

(a + b)³ = a³ + 3a²b + 3ab² + b³

Example: Expand (x + 2)³.

示例：展开 (x + 2)³。

Let a = x and b = 2:

令 a = x，b = 2：

(x + 2)³ = x³ + 3x²(2) + 3x(2)² + 2³ = x³ + 6x² + 12x + 8

Pascal’s triangle is especially useful for higher powers, such as (2x – 1)⁵, where direct multiplication would be extremely time-consuming.

帕斯卡三角对于高次幂尤其有用，例如 (2x – 1)⁵，如果直接相乘会非常耗时。

9. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Students often lose marks in exams because of small but repeated errors in bracket expansion. The most common mistakes are listed below.

学生在考试中常因括号展开时的小错误而丢分。下面是最高频的错误。

Multiplying only the first term inside the bracket. In 3(2x + 7), some students write 6x + 7 instead of 6x + 21. Always multiply every term inside the bracket.

只乘括号里的第一项。例如在 3(2x + 7) 中，有些学生写成 6x + 7 而不是 6x + 21。永远要把括号里的每一项都乘到。
Losing negative signs. The expression – (x – 4) should become -x + 4, not -x – 4. Track signs carefully when the outside term is negative.

丢失负号。表达式 – (x – 4) 应变为 -x + 4，而不是 -x – 4。当外面是负数时要仔细追踪符号。
Forgetting the middle term in a perfect square. (x + 3)² is x² + 6x + 9, not x² + 9. The term 2ab must always appear.

忘记完全平方的中间项。(x + 3)² 应为 x² + 6x + 9，而不是 x² + 9。中间项 2ab 必须出现。
Not collecting like terms after FOIL. After writing the four products, always check whether any terms can be combined. For example, 10x and 3x in a previous section become 13x.

FOIL 后没有合并同类项。写出四个乘积后，务必检查是否有可以合并的项。例如前面例子中的 10x 与 3x 合并为 13x。
Misapplying the square to only one factor. In (2x)², the answer is 4x², not 2x². Square the coefficient as well as the variable.

只对其中一部分平方。在 (2x)² 中，答案是 4x²，而不是 2x²。平方时系数和变量都要平方。

10. Practice Questions | 练习

Try the following questions on your own before checking the answers. Expanding brackets accurately requires practice, and these questions cover the most important patterns.

请先独立思考完成以下题目，再对照答案。熟练展开括号需要练习，这些题目覆盖了最重要的题型模式。

1. Expand 4(3x – 7).

1. 展开 4(3x – 7)。
2. Expand -2x(5x² – 3x + 2).

2. 展开 -2x(5x² – 3x + 2)。
3. Expand (x + 6)(x – 4).

3. 展开 (x + 6)(x – 4)。
4. Expand (2x – 5)(3x + 2).

4. 展开 (2x – 5)(3x + 2)。
5. Expand (x + 7)².

5. 展开 (x + 7)²。
6. Expand (3x – 2)(3x + 2).

6. 展开 (3x – 2)(3x + 2)。
7. Expand (x – 1)³.

7. 展开 (x – 1)³。

Answers:

答案：

1. 12x – 28.

1. 12x – 28。
2. -10x³ + 6x² – 4x.

2. -10x³ + 6x² – 4x。
3. x² + 2x – 24.

3. x² + 2x – 24。
4. 6x² – 11x – 10.

4. 6x² – 11x – 10。
5. x² + 14x + 49.

5. x² + 14x + 49。
6. 9x² – 4.

6. 9x² – 4。
7. x³ – 3x² + 3x – 1.

7. x³ – 3x² + 3x – 1。

If you made any errors, go back to the relevant section and identify exactly which step caused the mistake. This is the fastest way to improve.

如果出现错误，请回到对应小节，找出具体是哪一步导致了问题。这是提高最快的方法。

11. Summary | 总结

Expanding brackets is a core algebraic operation that you will use throughout A-Level Mathematics. The key rules are simple, but they must be applied carefully.

括号展开是 A-Level 数学中贯穿始终的核心代数运算。关键规则虽然简单，但必须谨慎应用。

For a single bracket, multiply the outside term by every term inside the bracket.

对于单括号，将外面项乘以括号内的每一项。
For two brackets, multiply every term in the first bracket by every term in the second bracket.

对于两个括号，将第一个括号中的每一项与第二个括号中的每一项相乘。
Use FOIL only as an organising tool for two binomials; it is not a separate rule.

FOIL 只是两个二项式相乘时的组织工具，并不是一条独立规则。
Remember the special identities: (a + b)² = a² + 2ab + b² and (a + b)(a – b) = a² – b².

记住特殊恒等式：(a + b)² = a² + 2ab + b² 以及 (a + b)(a – b) = a² – b²。
For higher powers, use Pascal’s triangle or expand step by step from left to right.

对于高次幂，可以使用帕斯卡三角，或者从左到右逐步展开。

Bracket expansion is not just a mechanical process; every step follows directly from the distributive law. Once you understand why the method works, you will be able to apply it to new situations with confidence.

括号展开不只是机械运算，每一步都直接来源于分配律。一旦你理解了方法背后的原理，就能自信地把该技能应用到各种新情境中。

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A-Level Maths: Expansion of Brackets in Algebra | A-Level 数学：括号展开的代数运算