Expanding Brackets and Collecting Like Terms | 去括号与合并同类项

📚 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

Take extra care when a minus sign appears before 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。

Also 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

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Expanding Brackets and Collecting Like Terms | 去括号与合并同类项