📚 Mastering Algebraic Expressions: Expanding and Simplifying | 掌握代数表达式：展开与化简
Algebra is one of the most important building blocks in the Cambridge KS3 mathematics curriculum. In many textbook exercises, including practice questions such as the one on page 68, question 2, you are asked to expand brackets and simplify expressions. This article breaks down the key skills step by step so you can work confidently and accurately.
代数是剑桥 KS3 数学课程中最重要的基础模块之一。在许多教材练习中，包括第 68 页第 2 题这样的题目，都会要求你展开括号并化简表达式。本文将逐步拆解关键技能，帮助你自信、准确地解题。
1. What Is an Algebraic Expression? | 什么是代数表达式？
An algebraic expression is a combination of numbers, letters and operation symbols. For example, 3x + 5, 2a − 7 and 4(x + 3) are all algebraic expressions. Unlike an equation, an expression does not contain an equals sign.
代数表达式是由数字、字母和运算符号组成的式子。例如 3x + 5、2a − 7 和 4(x + 3) 都是代数表达式。与方程不同，表达式不包含等号。
In KS3 work, the letter usually stands for an unknown or changing value. The expression tells you how to combine that value with the given numbers. Understanding this idea is the first step before you learn to expand brackets.
在 KS3 阶段，字母通常代表未知数或变化的量。表达式说明如何将该数值与给定数字进行组合。理解这一点是学习展开括号前的第一步。
2. Key Terms: Variables, Coefficients and Constants | 关键术语：变量、系数与常数
You need to know three important words before expanding brackets. A variable is the letter, such as x or y. A coefficient is the number in front of a variable, such as the 3 in 3x. A constant is a number on its own, such as 5.
在展开括号之前，你需要掌握三个重要术语。变量是字母，如 x 或 y。系数是变量前面的数字，例如 3x 中的 3。常数是单独出现的数字，例如 5。
In the expression 3x + 5 − 2y, the variables are x and y, the coefficients are 3 and −2, and the constant is 5. Identifying these parts helps you simplify correctly.
在表达式 3x + 5 − 2y 中，变量是 x 和 y，系数是 3 和 −2，常数是 5。识别这些部分有助于正确化简。
| Expression | Variable(s) | Coefficient(s) | Constant |
|---|---|---|---|
| 3x + 5 − 2y | x, y | 3, −2 | 5 |
| 4a − 7 | a | 4 | −7 |
| −2x + 3y + 1 | x, y | −2, 3 | 1 |
3. Why Expanding Brackets Matters | 为什么展开括号很重要
Expanding brackets helps you rewrite an expression in a simpler or more useful form. It is used in solving equations, working with formulas and later in quadratic graphs. Cambridge KS3 tests often check whether you can expand correctly before moving on to harder topics.
展开括号有助于将表达式改写成更简单或更有用的形式。它常用于解方程、处理公式以及后续的二次函数图像。剑桥 KS3 测试经常会考察你是否能正确展开，然后才进入更难的主题。
For example, 3(x + 4) looks compact, but after expanding it becomes 3x + 12. This expanded form makes it much easier to combine with other terms in the same question.
例如，3(x + 4) 看起来紧凑，但展开后变成 3x + 12。这种展开形式使你更容易将它与同一题中的其他项进行合并。
4. Collecting Like Terms | 合并同类项
Like terms contain exactly the same variable raised to the same power. For example, 3x and 5x are like terms, but 3x and 3x² are not like terms because the powers differ.
同类项包含完全相同的变量且变量的指数相同。例如 3x 和 5x 是同类项，但 3x 和 3x² 不是同类项，因为指数不同。
To collect like terms, add or subtract the coefficients and keep the variable unchanged. For example, 2x + 5x − 3x simplifies to 4x. This skill is essential after expanding brackets.
合并同类项时，将系数相加或相减，变量保持不变。例如 2x + 5x − 3x 化简为 4x。这是展开括号后必不可少的技能。
2x + 5x − 3x = 4x
Numbers without variables also collect together. For example, 6 + 4 − 3 simplifies to 7. Always keep the variable terms and the constant terms separate when you simplify.
不含变量的数字也要合并。例如 6 + 4 − 3 化简为 7。化简时始终将含变量的项和常数项分开。
5. Expanding a Single Bracket | 展开单个括号
he term outside. The rule is a(b + c) = ab + ac. For example, 3(x + 4) expands to 3x + 12.
展开是指用括号外的项乘以括号内的每一项。规则是 a(b + c) = ab + ac。例如 3(x + 4) 展开后为 3x + 12。
ltiply the sign as well. For example, −2(x − 5) expands to −2x + 10 because −2 × −5 = +10.
如果括号外是负号，记得连同符号一起相乘。例如 −2(x − 5) 展开为 −2x + 10，因为 −2 × −5 = +10。
3(x + 4) = 3x + 12
−2(x − 5) = −2x + 10
Always write the multiplication line clearly. If the outside term is a number, multiply it by every term inside. If the outside term is a variable, such as in x(x + 2), the result is x² + 2x.
务必清楚写出乘法步骤。如果括号外是数字，将其与括号内每一项相乘。如果括号外是变量，例如 x(x + 2)，结果是 x² + 2x。
6. Expanding and Simplifying: Two-Step Problems | 展开并化简：两步题
Many Cambridge KS3 questions ask you to expand first and then collect like terms. For example, expand 2(x + 3) + 4(x − 1). Start by expanding each bracket separately, then combine the x terms and the constants.
许多剑桥 KS3 题目要求先展开再合并同类项。例如展开 2(x + 3) + 4(x − 1)。首先分别展开每个括号，然后合并 x 项和常数项。
The expansion gives 2x + 6 + 4x − 4. Collecting like terms gives 6x + 2. Always show both steps clearly in your working.
展开得到 2x + 6 + 4x − 4。合并同类项得到 6x + 2。解题过程中务必清晰展示这两步。
2(x + 3) + 4(x − 1) = 2x + 6 + 4x − 4 = 6x + 2
When there are subtraction signs, be extra careful. For example, 5(x + 2) − 3(x − 1) expands to 5x + 10 − 3x + 3, which simplifies to 2x + 13.
当出现减号时，要格外小心。例如 5(x + 2) − 3(x − 1) 展开为 5x + 10 − 3x + 3，化简为 2x + 13。
7. Expanding Double Brackets | 展开双括号
To expand two brackets such as (x + 2)(x + 5), multiply each term in the first bracket by each term in the second bracket. This is sometimes called the FOIL method: First, Outer, Inner, Last.
要展开两个括号，如 (x + 2)(x + 5)，将第一个括号中的每一项分别乘以第二个括号中的每一项。这种方法有时称为 FOIL 法：首项、外项、内项、末项。
uter x × 5 = 5x, Inner 2 × x = 2x, Last 2 × 5 = 10. Then collect like terms: x² + 7x + 10.
按以下方式相乘：首项 x × x = x²，外项 x × 5 = 5x，内项 2 × x = 2x，末项 2 × 5 = 10。然后合并同类项：x² + 7x + 10。
(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10
For double brackets with subtraction, such as (x − 3)(x + 2), the same method applies. You get x² + 2x − 3x − 6, which simplifies to x² − x − 6.
对于含有减法的双括号，如 (x − 3)(x + 2)，使用相同的方法。你会得到 x² + 2x − 3x − 6，化简为 x² − x − 6。
8. Sign Rules When Expanding | 展开时的符号规则
× negative = negative, negative × positive = negative, and negative × negative = positive.
符号错误在 KS3 代数中非常常见。记住以下规则：正 × 正 = 正，正 × 负 = 负，负 × 正 = 负，负 × 负 = 正。
+ × + = +
+ × + = +
+ × − = −
+ × − = −
− × + = −
− × + = −
− × − = +
− × − = +
+ 2), the term −3 × x gives −3x and −3 × +2 gives −6. The final expression is x² − x − 6.
例如，在 (x − 3)(x + 2) 中，−3 × x 得到 −3x，−3 × +2 得到 −6。最终表达式为 x² − x − 6。
(x − 3)(x + 2) = x² + 2x − 3x − 6 = x² − x − 6
to every multiplication. For example, −4(x − 3) means −4 × x and −4 × −3, giving −4x + 12.
当括号外面是负项
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