Simplifying Algebraic Expressions: Collecting Like Terms, Expanding Brackets and Factorising | 代数式化简：合并同类项、去括号与因式分解

📚 Simplifying Algebraic Expressions: Collecting Like Terms, Expanding Brackets and Factorising | 代数式化简：合并同类项、去括号与因式分解

Algebraic simplification is a core skill in the Cambridge Lower Secondary Mathematics course. It allows you to rewrite expressions in a shorter or more useful form without changing their value. This article covers the essential techniques of collecting like terms, expanding brackets and factorising, with worked examples and common exam tips.

代数式化简是剑桥初中数学课程中的核心技能。它可以让你在不改变原式数值的前提下，把表达式改写成更简短或更有用的形式。本文涵盖合并同类项、去括号以及因式分解等关键技巧，并提供典型例题与常见考试建议。

1. Algebraic Language and Key Terms | 代数语言与关键术语

An algebraic expression is made up of numbers, letters and operations such as addition, subtraction, multiplication and division. Letters are called variables because their values can change, while a number on its own is called a constant.

代数式由数、字母以及加、减、乘、除等运算组成。字母称为变量，因为它们的值可以变化；单独出现的数字则称为常数项。

In the expression 4x² + 3x − 7, the parts 4x², 3x and −7 are called terms. The number in front of a variable is the coefficient, so the coefficient of x² is 4 and the coefficient of x is 3.

在表达式 4x² + 3x − 7 中，4x²、3x 和 −7 都称为项。变量前面的数称为系数，因此 x² 的系数是 4，x 的系数是 3。

Like terms have exactly the same variables raised to the same powers, such as 3a and −5a.
同类项必须含有完全相同的变量且变量的指数相同，例如 3a 与 −5a。
Unlike terms have different variable parts, such as 2x and 2x², or 3ab and 4a.
不同类项则变量部分不同，例如 2x 与 2x²，或 3ab 与 4a。

Remember that only like terms can be combined by addition or subtraction. Writing terms in alphabetical order and descending powers helps you spot like terms quickly.

请记住，只有同类项才能通过加法或减法合并。按照字母顺序和降幂排列各项，可以帮助你更快地找出同类项。

2. Collecting Like Terms | 合并同类项

To simplify an expression, first identify the like terms and then add or subtract their coefficients. The variable part stays unchanged.

化简表达式时，首先要找出所有同类项，然后将它们的系数相加或相减，变量部分保持不变。

3a + 5a = 8a

Here, 3a and 5a are like terms because both contain the variable a. Adding their coefficients gives 3 + 5 = 8, so the result is 8a.

这里 3a 和 5a 是同类项，因为都含有变量 a。将系数相加得到 3 + 5 = 8，因此结果是 8a。

7b − 2b = 5b

Subtraction works in the same way. The coefficient of b is 7 − 2 = 5, so the simplified term is 5b.

减法同样如此。b 的系数为 7 − 2 = 5，所以化简后的项是 5b。

2x + 3y + x − y = 3x + 2y

Collect the x terms together and the y terms together. x terms: 2x + x = 3x. y terms: 3y − y = 2y. The final expression is 3x + 2y.

把含有 x 的项和含有 y 的项分别合并。x 项：2x + x = 3x。y 项：3y − y = 2y。最终表达式为 3x + 2y。

In more complex examples, list each group of like terms before combining. This reduces mistakes when many negative signs are involved.

在更复杂的例子中，可以先列出每一组同类项再进行合并。当出现大量负号时，这种做法能减少错误。

3. Collecting Like Terms with Negative Coefficients | 含负系数的同类项合并

Negative coefficients require careful attention. A common method is to think of each term as having a sign attached to it, such as +4x or −2x.

负系数需要格外小心。一种常用方法是把每一项都看作带有自己的符号，例如 +4x 或 −2x。

4x − 3 − 2x + 5 = 2x + 2

Group the x terms: 4x − 2x = 2x. Group the constants: −3 + 5 = 2. The expression simplifies to 2x + 2.

合并 x 项：4x − 2x = 2x。合并常数项：−3 + 5 = 2。表达式化简为 2x + 2。

−2a + 3b − 5a − b = −7a + 2b

For the a terms, −2a − 5a gives −7a. For the b terms, 3b − b gives 2b. Therefore the simplified form is −7a + 2b.

对于 a 项，−2a − 5a 得到 −7a。对于 b 项，3b − b 得到 2b。因此化简结果是 −7a + 2b。

When collecting constants, always use the sign directly before the number. This is especially important in subtraction chains such as 6 − 3 + 8, which equals 11, not 1.

合并常数项时，一定要使用数字前面的符号。这一点在 6 − 3 + 8 这样的连续加减中尤其重要，它等于 11，而不是 1。

4. Expanding a Single Bracket | 单项式乘括号

Expanding brackets means removing the brackets by multiplying the term outside by every term inside the bracket. This uses the distributive law.

去括号是指将括号外的项与括号内的每一项相乘，从而去掉括号。这运用了分配律。

a(b + c) = ab + ac

Every term inside the bracket must be multiplied by the outside term. Do not multiply only the first term and forget the second one.

括号内的每一项都必须与括号外的项相乘。不要只乘第一项而忘记乘第二项。

3(x + 4) = 3x + 12

Multiply 3 by x to get 3x, then multiply 3 by 4 to get 12. The expanded form is 3x + 12.

用 3 乘 x 得到 3x，再用 3 乘 4 得到 12。展开后的式子是 3x + 12。

−2(3y − 5) = −6y + 10

When the outside term is negative, remember to multiply the sign as well. −2 × 3y = −6y and −2 × −5 = +10.

当括号外的项为负数时，要同时考虑符号。−2 × 3y = −6y，而 −2 × −5 = +10。

You can also expand where the outside term is a variable, such as x(x − 7) = x² − 7x. The same rule applies.

括号外的项也可以是变量，例如 x(x − 7) = x² − 7x。规则仍然相同。

5. Expanding and Simplifying | 去括号后化简

Many exam questions ask you to expand two or more brackets and then simplify the result by collecting like terms. Work step by step to avoid losing terms.

许多考试题要求先展开两个或更多括号，再合并同类项进行化简。要一步一步进行，避免漏项。

3(2a + 1) + 4(a − 2) = 6a + 3 + 4a − 8 = 10a − 5

First expand 3(2a + 1) to get 6a + 3. Then expand 4(a − 2) to get 4a − 8. Finally collect like terms: 6a + 4a = 10a and 3 − 8 = −5.

先把 3(2a + 1) 展开为 6a + 3。再把 4(a − 2) 展开为 4a − 8。最后合并同类项：6a + 4a = 10a，3 − 8 = −5。

5(x − 2) − 2(3x + 1) = 5x − 10 − 6x − 2 = −x − 12

This example includes a subtraction between the two brackets. Expand the second bracket first, giving −2(3x + 1) = −6x − 2. Then collect terms: 5x − 6x = −x and −10 − 2 = −12.

这个例子中两个括号之间是减法。先展开第二个括号，得到 −2(3x + 1) = −6x − 2。然后合并同类项：5x − 6x = −x，−10 − 2 = −12。

Writing the intermediate expansion line is very useful in tests because examiners can see where any arithmetic slip occurs.

在考试中写出中间展开步骤非常有用，因为阅卷者可以看到任何计算失误发生在哪里。

6. Expanding Double Brackets | 双括号展开

When two binomials are multiplied, every term in the first bracket must be multiplied by every term in the second bracket. The result is often called a quadratic expression.

当两个二项式相乘时，第一个括号中的每一项都要与第二个括号中的每一项相乘。所得结果通常称为二次式。

(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

Multiply x by both terms in the second bracket: x² + 3x. Then multiply 2 by both terms: 2x + 6. Finally add the like terms 3x and 2x to get 5x.

先用 x 乘第二个括号中的两项：x² + 3x。再用 2 乘这两项：2x + 6。最后合并同类项 3x 与 2x，得到 5x。

(x − 3)(x + 5) = x² + 5x − 3x − 15 = x² + 2x − 15

Watch the signs carefully: −3 × x = −3x and −3 × 5 = −15. The middle terms 5x and −3x combine to give 2x.

注意符号：−3 × x = −3x，−3 × 5 = −15。中间项 5x 与 −3x 合并后得到 2x。

(x + 4)² = (x + 4)(x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16

A common error is to write (x + 4)² as x² + 16. The correct expansion must include the middle term 8x.

一个常见错误是把 (x + 4)² 写成 x² + 16。正确的展开必须包含中间项 8x。

7. Factorising by Taking Out Common Factors | 提取公因式

Factorising is the reverse of expanding. It means rewriting an expression as a product of brackets by identifying the highest common factor of all terms.

因式分解是去括号的逆运算。它通过找出所有项的最高公因式，将表达式改写为括号的乘积形式。

6x + 9 = 3(2x + 3)

The highest common factor of 6x and 9 is 3. Divide each term by 3: 6x ÷ 3 = 2x and 9 ÷ 3 = 3. Then write 3 outside the bracket.

6x 和 9 的最高公因式是 3。将每一项除以 3：6x ÷ 3 = 2x，9 ÷ 3 = 3。然后把 3 写在括号外。

10a² − 5a = 5a(2a − 1)

Here the common factor is 5a because both terms contain the variable a. 10a² ÷ 5a = 2a and −5a ÷ 5a = −1.

这里公因式是 5a，因为两项都含有变量 a。10a² ÷ 5a = 2a，−5a ÷ 5a = −1。

Always check your factorisation by expanding the brackets again. The expanded form should match the original expression exactly.

因式分解后一定要重新展开括号来检查。展开后的形式必须与原式完全一致。

8. Factorising Simple Quadratic Expressions | 简单二次式的因式分解

A quadratic expression of the form x² + bx + c can often be factorised into two brackets (x + p)(x + q), where p and q multiply to give c and add to give b.

形如 x² + bx + c 的二次式通常可以分解为两个括号 (x + p)(x + q)，其中 p 与 q 的乘积为 c，和为 b。

x² + 6x + 8 = (x + 2)(x + 4)

We need two numbers that multiply to 8 and add to 6. The numbers 2 and 4 work because 2 × 4 = 8 and 2 + 4 = 6.

我们需要两个数，乘积为 8，和为 6。2 和 4 满足条件，因为 2 × 4 = 8，2 + 4 = 6。

x² − 5x + 6 = (x − 2)(x − 3)

The constant 6 is positive, so both numbers have the same sign. Because the middle term is −5x, both numbers are negative: −2 × −3 = 6 and −2 + −3 = −5.

常数项 6 为正，因此两个数同号。由于中间项是 −5x，两个数都为负数：−2 × −3 = 6，−2 + −3 = −5。

x² − 9 = (x + 3)(x − 3)

This is called the difference of two squares. There is no x term because the two middle terms +3x and −3x cancel each other out.

这称为平方差。没有 x 项是因为中间项 +3x 与 −3x 相互抵消。

9. Applying Algebra to Geometry | 代数在几何中的应用

Simplifying algebraic expressions is often tested in context, for example when finding the perimeter or area of a shape with side lengths expressed in algebra.

代数式化简常常出现在实际情境中，例如求边长为代数式的图形的周长或面积。

Perimeter = 2(2x + 3) + 2(x + 1) = 4x + 6 + 2x + 2 = 6x + 8

For a rectangle with length 2x + 3 and width x + 1, expand and simplify to find the perimeter expression 6x + 8.

对于长为 2x + 3、宽为 x + 1 的长方形，展开并化简后得到周长表达式 6x + 8。

Area = (1/2)(x)(x + 4) = (1/2)(x² + 4x) = x²/2 + 2x

For a triangle with base x and height x + 4, use the formula Area = 1/2 × base × height. Multiply x by x + 4, then divide each term by 2.

对于底为 x、高为 x + 4 的三角形，使用面积公式“面积 = 1/2 × 底 × 高”。先将 x 与 x + 4 相乘，再把每一项除以 2。

When substituting values, simplify the algebraic expression first. It makes the arithmetic easier and reduces mistakes, especially when x is negative.

当代入数值时，应先把代数式化简。这样计算更简单，也能减少错误，尤其是当 x 为负数时。

10. Common Errors and Exam Tips | 常见错误与考试建议

Being aware of common mistakes can significantly improve your accuracy in algebra questions. The table below summarises frequent errors and the correct approach.

了解常见错误可以显著提高代数题的准确性。下表总结了常见错误及正确做法。

Common error	Correct approach
(x + 4)² = x² + 16	(x + 4)² = x² + 8x + 16, because the middle term cannot be omitted.
Losing the negative sign when expanding −2(3y − 5)	Multiply −2 by −5 to get +10, so the result is −6y + 10.
Combining 2x and 2x² as 4x²	2x and 2x² are unlike terms and cannot be added.
Forgetting to factorise fully: 6x + 12 = 3(2x + 4)	The fully factorised form is 6(x + 2), using the highest common factor 6.

Always write each step clearly, especially the expanded line before collecting like terms. This earns method marks even if a small arithmetic error is made later.

每一步都要写清楚，尤其是先写出展开后的式子再合并同类项。这样即使后面出现小的计算错误，也能获得步骤分。

A short final check is to substitute a simple value for x, such as x = 1, into the original and simplified expressions. If both give the same result, your simplification is likely correct.

最后可以用一个简单的数值（例如 x = 1）代入原式和化简后的式子进行检验。如果两者结果相同，化简很可能是正确的。

11. Summary | 小结

Simplifying algebraic expressions requires strong control of signs, coefficients and powers. The key techniques are collecting like terms, expanding brackets using the distributive law, and factorising by taking out common factors or by finding two numbers that fit a quadratic pattern.

化简代数式需要熟练掌握符号、系数和指数。关键技巧包括合并同类项、运用分配律去括号，以及通过提取公因式或寻找满足二次式规律的两个数进行因式分解。

Practice these methods regularly and always check your work by expanding or substituting simple values. This will build speed and confidence for Cambridge KS3 mathematics assessments.

定期练习这些方法，并始终通过展开或代入简单数值来检查答案。这将提高速度与信心，为剑桥 KS3 数学测评做好准备。

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Simplifying Algebraic Expressions: Collecting Like Terms, Expanding Brackets and Factorising | 代数式化简：合并同类项、去括号与因式分解