📚 Collecting Like Terms: A Complete IGCSE & A-Level Algebra Guide | 合并同类项：IGCSE 与 A-Level 代数完全指南
In IGCSE and A-Level algebra, simplifying an expression almost always begins with collecting like terms. This single skill turns a long, untidy expression into a short one that is far easier to substitute into, factorise, differentiate or graph, and it is examined in almost every algebra paper.
在 IGCSE 与 A-Level 代数中，化简表达式几乎总是从合并同类项开始。这一项技能能把冗长凌乱的表达式变成简短的形式，从而更容易代入求值、因式分解、求导或作图，而且它几乎出现在每一份代数试卷中。
In this article you will learn how to identify like terms, combine their coefficients correctly, handle negative signs without slipping up, work with several variables and powers at once, expand brackets before collecting, check your final answer by substitution, and apply everything to exam-style questions. All of the original key points are kept, with extra worked examples and a practice set added.
在本文中，你将学会如何识别同类项、正确合并它们的系数、不出错地处理负号、同时处理多个变量与幂、先展开括号再合并、通过代入法检验最终答案，并把这一切应用到考试风格的题目中。原文的全部关键信息点都会保留，并补充更多例题与一组练习题。
1. What Are Like Terms? | 什么是同类项？
Like terms are terms that have exactly the same variable part, including the same powers. Only the coefficients, the numbers in front, may be different. If two terms have the same letters raised to the same powers, they can be added or subtracted into a single term.
同类项是指变量部分完全相同（包括相同次幂）的项。只有系数，也就是前面的数字，可以不同。如果两个项的字母相同、且字母的指数也相同，就可以把它们加减成一个单独的项。
6ab and -9ab are like terms because both contain the product ab, and the order ab = ba does not change that.
例如，3x 与 -5x 是同类项，因为两者都含有 x 的一次方。2x² 与 4x² 也是同类项，因为两者都含有 x²。6ab 与 -9ab 也是同类项，因为两者都含有乘积 ab，而 ab = ba 的顺序变化并不影响这一点。
However, 3x and 3x² are not like terms: the variable x has different powers. Likewise, 5ab and 5a are not like terms because the variable parts ab and a are different. Terms with no variable part at all, such as 7 and -3, are like terms with each other and are called constants.
但是，3x 与 3x² 不是同类项：变量 x 的次幂不同。同样，5ab 与 5a 也不是同类项，因为变量部分 ab 与 a 不同。完全没有变量部分的项，例如 7 与 -3，彼此之间是同类项，称为常数项。
| Like terms 同类项 | Not like terms 非同类项 |
|---|---|
| 3x and -5x | 3x and 3x² |
| 2x² and 4x² | 5ab and 5a |
| -4xy and 7xy | 3ab² and 3a²b |
| 6ab and -9ab | 4x and 4y |
| 11 and -3 | x²y and xy² |
2. Identifying the Variable Part and the Coefficient | 识别变量部分与系数
Every term is made of two pieces: a coefficient and a variable part. In the term 8x², the coefficient is 8 and the variable part is x². In -3ab, the coefficient is -3 and the variable part is ab. The coefficient is always the number, including its sign.
每一个项都由两部分组成：系数和变量部分。在 8x² 中，系数是 8，变量部分是 x²。在 -3ab 中，系数是 -3，变量部分是 ab。系数始终是那个数字，并且要带上它的符号。
. Students who forget these invisible 1s often write answers such as xy + xy = xy instead of 2xy.
要小心隐藏的系数。项 x 其实就是 1x，系数是 1；项 -y 其实就是 -1y，系数是 -1；项 xy 其实就是 1xy。忘记这些看不见的 1 的学生，常常会把 xy + xy 写成 xy，而不是正确的 2xy。
When you read an expression, split it into terms at the + and - signs, and let the sign travel with the term that follows it. For example, the expression -5x + 3y - 7 is made of the three terms -5x, +3y and -7. Writing the terms out this way before you do anything else removes most sign errors later.
阅读一个表达式时，用 + 和 - 号把它切开，并让符号跟着它后面那个项一起走。例如表达式 -5x + 3y - 7 由 -5x、+3y 和 -7 三个项组成。先这样把项写清楚，后面的符号错误就会少很多。
- 8x² → coefficient 8, variable part x² 系数 8，变量部分 x²
- -3ab → coefficient -3, variable part ab 系数 -3，变量部分 ab
- x → coefficient 1, variable part x 系数 1，变量部分 x
- -y → coefficient -1, variable part y 系数 -1，变量部分 y
- -7 → coefficient -7, no variable part (constant) 系数 -7，无变量部分（常数）
3. Combining Coefficients: The Core Rule | 合并系数：核心法则
Once you know two terms are like terms, you simply add or subtract their coefficients and keep the variable part unchanged. The general rule for a single variable x is shown below.
一旦确定两个项是同类项，你只需把它们的系数相加或相减，变量部分保持不变。关于单一变量 x 的一般法则如下。
ax + bx = (a + b)x
ax − bx = (a − b)x
) = (a + b)x. The variable part is a common factor, so it is not affected by the addition.
这样做的依据是分配律：ax + bx = x(a + b) = (a + b)x。变量部分是公因式，因此不受加减运算的影响。
The most important warning in this whole topic is that the variable part never changes. The power of x stays exactly the same.
整个主题中最重要的一条提醒是：变量部分永远不变。x 的次幂保持完全一致。
3x + 4x = 7x (not 7x²)
x² + x² = 2x² (not x⁴)
- 7y + 3y = 10y 同类项相加，系数 7 + 3 = 10
- 9m − 4m = 5m 系数 9 − 4 = 5
- -2x + 8x = 6x 系数 -2 + 8 = 6
- 4t + t = 5t 记住 t 的系数是 1
- 6p − 6p = 0 合并结果为零，该项可以完全去掉
4. Handling Negative Signs Correctly | 正确处理负号
Negative coefficients cause more lost marks than any other part of this topic. The safe method is to rewrite every subtraction as adding a negative, then add the coefficients as signed numbers.
负系数造成的失分比这个主题中其他任何部分都多。最稳妥的方法是把每一个减法都改写成加上一个负数，然后按有符号数相加的方式计算系数。
5a − 8a = 5a + (−8a) = −3a
3b − (−4b) = 3b + 4b = 7b
−6c − 2c = −6c + (−2c) = −8c
Notice the third line: a minus sign in front of a bracket changes every sign inside it. This is the same rule you meet when subtracting polynomials, and it is worth practising until it becomes automatic.
注意第三行：括号前面的减号会改变括号内每一个项的符号。这与多项式减法中的规则完全相同，值得反复练习直到变成条件反射。
- −5x + 2x = −3x 系数 −5 + 2 = −3
- 4y − 9y = −5y 结果可以为负
- −a − 3a = −4a a 的系数是 1
- 8k − 8k = 0 同类项互相抵消
- −2d + 7d − 3d = 2d 先合正项再合负项
5. Several Variables and Powers at Once | 同时处理多个变量与幂
Many exam expressions contain more than one variable and more than one power. The rule does not change: group terms that match exactly, then work through one group at a time. It often helps to underline or box each group before adding.
许多考题表达式含有不止一个变量和不止一个幂。规则并没有变化：把完全匹配的项分到一组，然后一组一组地处理。在相加之前用下划线或方框把每一组标出来，往往很有帮助。
Order of letters does not matter, so yx² and x²y are the same variable part and can be collected together. What does matter is the power attached to each letter.
字母的顺序不影响判断，因此 yx² 与 x²y 的变量部分相同，可以合并。真正重要的是每个字母所带的指数。
| Expression 表达式 | Simplified 化简结果 | Reason 理由 |
|---|---|---|
| 4x²y + 3x²y | 7x²y | same variable part x²y 变量部分相同 |
| 5ab − 9ab | −4ab | coefficients 5 − 9 = −4 系数相减 |
| −4xy + xy | −3xy | xy has coefficient 1 xy 的系数是 1 |
| 2x² + 5x² − x² | 6x² | 2 + 5 − 1 = 6 系数计算 |
| 3ab² + 2a²b | cannot be simplified 不能合并 | different variable parts 变量部分不同 |
A useful habit is to write the final answer with the highest power first, for example 3x² + 10x − 4 rather than 10x − 4 + 3x². Examiners accept either order, but a consistent order makes your working much easier to check.
一个有用的习惯是把最高次幂写在最前面，例如写成 3x² + 10x − 4，而不是 10x − 4 + 3x²。考官两种顺序都接受，但保持一致的顺序会让你的解题过程更容易检查。
6. Adding and Subtracting Polynomials | 多项式的加法与减法
A polynomial is simply an expression with several terms, such as 5x² + 3x − 4. Collecting like terms is the standard way to simplify one. Work term by term, matching powers of x.
多项式就是含有若干项的表达式，例如 5x² + 3x − 4。合并同类项是化简它的标准方法。逐项处理，把 x 的同次幂配成一组。
Simplify 5x² + 3x − 2x² + 7x − 4. Group the x² terms, the x terms and the constants.
化简 5x² + 3x − 2x² + 7x − 4。把 x² 项、x 项和常数项分别分组。
(5x² − 2x²) + (3x + 7x) + (−4) = 3x² + 10x − 4
When whole brackets are subtracted, remove the bracket and reverse every sign inside before collecting. This single step is where most marks are lost.
当整个括号被减去时，先去括号，并把括号内每一项的符号都反过来，然后再合并。这一步是失分最集中的地方。
(3x² + 5x − 2) − (x² − 4x + 6)
= 3x² + 5x − 2 − x² + 4x − 6 = 2x² + 9x − 8
Check the middle sign carefully: −(−4x) becomes +4x, and −(+6) becomes −6. If you are unsure, rewrite the subtraction as adding the negative of each term before you simplify.
仔细检查中间的符号：−(−4x) 变成 +4x，−(+6) 变成 −6。如果不确定，先把减法改写成加上每一项的相反数，再进行化简。
7. Expanding Brackets Before Collecting | 先展开括号再合并
If an expression contains brackets, expand them first and only then collect like terms. Trying to collect across a bracket is a common and costly mistake.
如果表达式中含有括号，要先展开括号，然后才合并同类项。试图跨过括号直接合并，是一个常见且代价很高的错误。
3(x + 4) + 2(x − 1) = 3x + 12 + 2x − 2 = 5x + 10
2(3x − 1) − (x − 5) = 6x − 2 − x + 5 = 5x + 3
2x(x + 3) − x² = 2x² + 6x − x² = x² + 6x
In the second example, the bracket (x − 5) is multiplied by −1, so both signs flip. In the third, multiplying 2x by x gives 2x², and multiplying 2x by 3 gives 6x; only the x² terms can then be collected.
在第二个例子中，括号 (x − 5) 被乘以 −1，因此两个符号都要变号。在第三个例子中，2x 乘 x 得 2x²，2x 乘 3 得 6x，之后只有 x² 项可以合并。
- Always expand fully before collecting 一定要先完整展开再合并
- Multiply every term inside the bracket 括号内每一项都要乘到
- Watch the sign when a bracket is subtracted 括号被减时注意变号
- Collect by matching variable part, not by position 按变量部分分组，而不是按位置
8. Checking Your Answer by Substitution | 用代入法检验答案
Because collecting like terms does not change the value of an expression, you can always test your answer by choosing a number for the variable and evaluating both the original and the simplified expression. If the two values agree, your simplification is almost certainly correct.
由于合并同类项不改变表达式的值，你总可以给变量取一个数，分别计算原表达式与化简后表达式的值来检验。如果两个值相同，你的化简几乎肯定是对的。
Test 5x² + 3x − 2x² + 7x − 4 = 3x² + 10x − 4 with x = 2.
用 x = 2 检验 5x² + 3x − 2x² + 7x − 4 = 3x² + 10x − 4。
Original: 5(4) + 3(2) − 2(4) + 7(2) − 4 = 20 + 6 − 8 + 14 − 4 = 28
Simplified: 3(4) + 10(2) − 4 = 12 + 20 − 4 = 28
such as x = −1 is even better, because it exposes sign errors that positive numbers hide.
两者都得 28，说明化简正确。用 x = 1 检验很快，但用 x = −1 这样的负数更好，因为负数能暴露正数掩盖的符号错误。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Most errors in this topic fall into a small number of patterns. Learn to recognise them and you will protect a lot of easy marks in the exam.
这个主题中的大多数错误都可以归为几类固定模式。学会识别它们，就能在考试中保住大量容易拿的分数。
| Wrong 错误 | Right 正确 | Why 原因 |
|---|---|---|
| 3x + 2 = 5x | 3x + 2 | 2 is a constant, not a like term 2 是常数，不是同类项 |
| 3x + 4x = 7x² | 3x + 4x = 7x | powers never change when collecting 合并时指数不变 |
| 5a − 8a = 3a | 5a − 8a = −3a | 8 is larger, so the result is negative 8 更大，结果为负 |
| xy + xy = xy | xy + xy = 2xy | hidden coefficient 1 隐藏系数为 1 |
| 2(x
广告 / Sponsored 【现货】剑桥国际高中教材 Cambridge IGCSE & O Level Additional Mathemati… 数学 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥198.0更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |