📚 Expanding Brackets and Collecting Like Terms | 去括号与合并同类项
In KS3 Cambridge Mathematics, expanding brackets and collecting like terms are essential algebra skills. They help you rewrite expressions in a simpler form and prepare you for solving equations, factorising, and working with formulas.
在 KS3 剑桥数学中，去括号与合并同类项是重要的代数技能。它们帮助你将表达式改写成更简单的形式，并为你学习解方程、因式分解和运用公式打下基础。
1. Why Algebra Matters | 为什么代数很重要
Algebra is a branch of mathematics that uses letters and symbols to represent numbers and relationships. It allows us to describe patterns, rules, and problems in a general way.
代数是数学的一个分支，它使用字母和符号来表示数字和关系。它使我们能够以通用的方式描述规律、规则和问题。
Being able to simplify expressions makes later topics such as solving equations, drawing graphs, and checking formulas much easier.
能够化简表达式会使以后的内容（如解方程、绘制图像和验证公式）变得更容易。
In Cambridge KS3, you will often meet expressions like 3(x + 4) and need to write them without brackets.
在剑桥 KS3 课程中，你会经常遇到像 3(x + 4) 这样的表达式，并且需要将它们写成不带括号的形式。
2. Key Vocabulary | 核心词汇
An expression is a combination of numbers, variables, and operations, such as 3x + 5 or 2a − 7.
表达式是数字、变量和运算的组合，例如 3x + 5 或 2a − 7。
A term is a single part of an expression separated by + or − signs; in 3x + 5, the terms are 3x and 5.
项是由加号或减号分隔的表达式的一部分；在 3x + 5 中，项是 3x 和 5。
A coefficient is the number part of a term, for example the coefficient of 3x is 3.
系数是项中的数字部分，例如 3x 的系数是 3。
| English term | 中文术语 | Example |
|---|---|---|
| Expression | 表达式 | 4x + 9 |
| Term | 项 | 4x, 9 |
| Coefficient | 系数 | 4 in 4x |
| Variable | 变量 | x |
| Like terms | 同类项 | 2x and 5x |
3. Like Terms and Unlike Terms | 同类项与不同类项
Like terms have exactly the same variable parts, so 2x and 5x are like terms, while 2x and 3y are unlike terms.
同类项有完全相同的变量部分，因此 2x 和 5x 是同类项，而 2x 和 3y 不是同类项。
You can add or subtract like terms by combining their coefficients: 2x + 5x = 7x.
你可以通过合并系数来加减同类项：2x + 5x = 7x。
You cannot combine unlike terms into a single term, so 2x + 3y stays as it is.
你不能将不同类项合并为一个项，所以 2x + 3y 保持不变。
- 4a + 7a = 11a
- 9b − 3b = 6b
- 2x + 3y + x = 3x + 3y
- 5m + 2n − 2m = 3m + 2n
Notice that x and x² are not like terms because the powers differ.
注意 x 和 x² 不是同类项，因为变量的次数不同。
4. The Distributive Law | 分配律
The distributive law states that a(b + c) = ab + ac.
分配律表明 a(b + c) = ab + ac。
This means the term outside the bracket multiplies every term inside the bracket.
这意味着括号外的项要乘以括号内的每一项。
3(x + 4) = 3x + 12
The distributive law works for subtraction too: a(b − c) = ab − ac.
分配律同样适用于减法：a(b − c) = ab − ac。
Understanding this law is the key to expanding brackets correctly.
理解这个定律是正确去括号的关键。
5. Expanding a Single Bracket | 展开单个括号
To expand 5(2x − 3), multiply 5 by 2x and then 5 by −3.
要展开 5(2x − 3)，先将 5 乘以 2x，再将 5 乘以 −3。
The result is 10x − 15.
结果是 10x − 15。
Expand 7(2a + 3b − 4) by multiplying 7 by each term inside the bracket.
展开 7(2a + 3b − 4)，将 7 分别乘以括号内的每一项。
First multiply 7 by 2a to get 14a, then multiply 7 by 3b to get 21b, and finally multiply 7 by −4 to get −28.
先将 7 乘以 2a 得到 14a，然后将 7 乘以 3b 得到 21b，最后将 7 乘以 −4 得到 −28。
7(2a + 3b − 4) = 14a + 21b − 28
Always check that you multiply every term inside the bracket by the factor outside.
始终检查括号内的每一项是否都乘以括号外的因数。
6. Expanding and Simplifying | 展开并化简
After expanding two or more brackets, you often have like terms that can be collected.
在展开两个或多个括号后，通常会有可以合并的同类项。
For example, expand 2(x + 3) + 4(x − 1) to get 2x + 6 + 4x − 4, then collect like terms to obtain 6x + 2.
例如，展开 2(x + 3) + 4(x − 1) 得到 2x + 6 + 4x − 4，然后合并同类项得到 6x + 2。
2(x + 3) + 4(x − 1) = 2x + 6 + 4x − 4 = 6x + 2
Write each step clearly so you do not lose track of the signs.
每一步都要写清楚，这样你就不会弄混符号。
Another example is 5(y + 2) − 2(y − 3). First expand to get 5y + 10 − 2y + 6, then simplify to 3y + 16.
另一个例子是 5(y + 2) − 2(y − 3)。首先展开得到 5y + 10 − 2y + 6，然后化简为 3y + 16。
5(y + 2) − 2(y − 3) = 5y + 10 − 2y + 6 = 3y + 16
7. Expanding with Negative Multipliers | 负数乘数展开
When the number outside the bracket is negative, you must multiply each term by a negative number.
当括号外的数是负数时，你必须用负数乘以每一项。
For example, −3(y − 2) = −3y + 6 because −3 × (−2) = +6.
例如，−3(y − 2) = −3y + 6，因为 −3 × (−2) = +6。
A common error is to write −3y − 6; remember that two negatives multiply to make a positive.
一个常见错误是写成 −3y − 6；记住两个负数相乘得正数。
−3(y − 2) = −3y + 6
a bracket, as in 10 − (x + 4). This means 10 − 1(x + 4), so it becomes 10 − x − 4 = 6 − x.
当括号前出现减号时要格外小心，例如 10 − (x + 4)。这表示 10 − 1(x + 4)，因此它变为 10 − x − 4 = 6 − x。
8. Expanding Double Brackets | 展开双括号
To expand two binomials such as (x + 2)(x + 5), multiply each term in the first bracket by each term in the second bracket.
要展开两个二项式，例如 (x + 2)(x + 5)，用第一个括号中的每一项乘以第二个括号中的每一项。
You get x² + 5x + 2x + 10, which simplifies to x² + 7x + 10.
你得到 x² + 5x + 2x + 10，化简为 x² + 7x + 10。
This method is sometimes called FOIL: First, Outer, Inner, Last.
这种方法有时称为 FOIL：首项、外项、内项、末项。
(x + 2)(x + 5) = x² + 7x + 10
The general pattern for two binomials is (a + b)(c + d) = ac + ad + bc + bd.
两个二项式的一般模式是 (a + b)(c + d) = ac + ad + bc + bd。
For example, (y + 3)(y − 2) = y² − 2y + 3y − 6 = y² + y − 6.
例如，(y + 3)(y − 2) = y² − 2y + 3y − 6 = y² + y − 6。
9. Common Mistakes | 常见错误
Forgetting to multiply every term inside the bracket is one of the most common mistakes.
忘记乘以括号内的每一项是最常见的错误之一。
Another mistake is mishandling negative signs when expanding; always apply the sign of the multiplier carefully.
另一个错误是在展开时处理负号不当；务必仔细应用乘数的符号。
Do not try to combine unlike terms such as x² and x.
不要试图合并不同类项，例如 x² 和 x。
lso avoid writing (x + 2)² as x² + 4; it must be expanded as (x + 2)(x + 2) = x² + 4x + 4.
还要避免将 (x + 2)² 写成 x² + 4；它必须展开为 (x + 2)(x + 2) = x² + 4x + 4。
10. Word Problems and Real-Life Links | 应用题与现实联系
Word problems often describe a situation that can be written as an expression with brackets.
应用题通常描述一种
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