📚 Collecting Like Terms: The Complete Guide | 合并同类项完全指南
In algebra, many expressions look complicated, but most of them can be made much simpler by collecting like terms. This topic is one of the most important foundation skills in IGCSE Mathematics because it is used in solving equations, plotting graphs, handling formulas and answering word problems.
在代数中，许多表达式看起来很复杂，但大多数都可以通过合并同类项来化简。这是 IGCSE 数学中最重要的基础技能之一，因为它广泛应用于解方程、绘制图像、处理公式以及解答应用题。
This article explains what like terms are, how to collect them correctly, how to handle negative signs, powers and brackets, and how to avoid the mistakes that examiners see every year.
本文将解释什么是同类项、如何正确合并同类项、如何处理负号、幂次与括号，以及如何避免考官每年都会见到的那些错误。
1. What Are Algebraic Terms? | 什么是代数项
In algebra, a term is a single number, a single letter, or a product of numbers and letters. Terms are separated from each other by addition or subtraction signs.
在代数中，项是单个数字、单个字母，或数字与字母的乘积。项与项之间由加号或减号分隔。
For example, in the expression 5x + 2y − 3, the three terms are 5x, 2y and −3.
例如，在表达式 5x + 2y − 3 中，三项分别是 5x、2y 和 −3。
A term may contain a coefficient multiplied by a power, such as 7x² or −4ab, and the sign written in front of a term always belongs to that term.
一个项可以包含系数与幂的乘积，例如 7x² 或 −4ab，而写在项前面的符号永远属于该项本身。
5x + 2y − 3 → three terms: 5x, +2y, −3
Nothing can be simplified yet, because no two of these terms are alike. The first skill, therefore, is to be able to break an expression into its separate terms correctly.
此时还无法化简，因为这三项互不相同。因此，第一项技能就是能够正确地把一个表达式拆分成单独的项。
2. Coefficients, Variable Parts and Constants | 系数、变量部分与常数项
The number part of a term is called the coefficient, and the letter part is called the variable part. In 5x, the coefficient is 5 and the variable part is x.
项中的数字部分称为系数，字母部分称为变量部分。在 5x 中，系数是 5，变量部分是 x。
If a term is written with no visible number, such as x or −y, the coefficient is understood to be 1 or −1. Writing 1x is correct but unnecessary, so mathematicians simply write x.
如果某一项没有明显写出数字，例如 x 或 −y，则系数默认为 1 或 −1。写成 1x 并没有错，但没必要，所以通常直接写作 x。
A term with no letter at all, such as −3 or 8, is called a constant term, because its value never changes.
完全不含字母的项，例如 −3 或 8，称为常数项，因为它的值永远不变。
| Term | 项 | Coefficient | 系数 | Variable part | 变量部分 | Type | 类型 |
|---|---|---|---|
| 7x² | 7 | x² | variable term | 变量项 |
| −4ab | −4 | ab | variable term | 变量项 |
| x | 1 | x | variable term | 变量项 |
| −3 | −3 | none | 无 | constant | 常数项 |
Being able to name the parts of a term makes the next step much easier, because collecting like terms is really only about adding and subtracting coefficients.
能够准确说出项的各个部分，会让下一步变得容易得多，因为合并同类项本质上是加减系数而已。
3. Recognising Like Terms | 识别同类项
Like terms have exactly the same variable part, including the same letters and the same powers. Only the coefficients may be different.
同类项具有完全相同的变量部分，包括相同的字母和相同的幂次，只有系数可以不同。
For example, 4x and −7x are like terms, because both have variable part x. However, 3x and 3x² are not like terms, because the powers of x are different.
例如，4x 和 −7x 是同类项，因为它们的变量部分都是 x。而 3x 和 3x² 不是同类项，因为 x 的幂次不同。
| Pair | 一组项 | Like terms? | 是否同类项 | Reason | 原因 |
|---|---|---|
| 6a and −2a | Yes | 是 | same letter, same power | 字母相同、幂次相同 |
| 5x² and 9x² | Yes | 是 | both x squared | 都是 x 的平方 |
| 3x and 3x² | No | 否 | different powers | 幂次不同 |
| 4ab and 4a | No | 否 | different letters | 字母不同 |
| 7 and −2 | Yes | 是 | both constants | 都是常数项 |
Notice that the letters in a term are usually written in alphabetical order, so ba is normally written as ab. This makes it easier to spot like terms in a long expression.
注意，项中的字母通常按字母表顺序书写，因此 ba 通常写作 ab。这样在冗长的表达式中更容易发现同类项。
4. The Golden Rule of Collecting Like Terms | 合并同类项的黄金法则
The golden rule is simple: when you collect like terms, the variable part stays exactly the same and only the coefficients are added or subtracted.
黄金法则很简单：合并同类项时，变量部分保持完全不变，只对系数进行加减。
A useful way to picture this is to think of x as an object. If 4 apples plus 7 apples gives 11 apples, then 4x plus 7x gives 11x.
一个有用的想象方式是把 x 想成某种物品。如果 4 个苹果加 7 个苹果等于 11 个苹果，那么 4x 加 7x 就等于 11x。
4x + 7x = (4 + 7)x = 11x
33, and the right side is 11 × 3 = 33. Both sides agree, so the simplification is correct.
我们总可以用代入数字的方法检验结果。令 x = 3：左边是 4 × 3 + 7 × 3 = 12 + 21 = 33，右边是 11 × 3 = 33。两边相等，说明化简正确。
ame rule works for powers and for products of letters: 6x² + 2x² = 8x² and 5ab − ab = 4ab.
同样的规则也适用于幂和字母乘积：6x² + 2x² = 8x²，5ab − ab = 4ab。
5. A Reliable Step-by-Step Method | 可靠的分步方法
Long expressions are easy to get wrong, so it is worth using the same four steps every time.
冗长的表达式很容易出错，因此每次都使用同样的四个步骤是值得的。
Step 1: Rewrite the expression and identify every term, including its sign.
第一步：重新抄写表达式，找出每一个项，包括它前面的符号。
Step 2: Group the like terms together, for example by underlining or circling each family of terms.
第二步：把同类项归到一起，例如用下划线或圈出每一"家族"的项。
Step 3: Add or subtract the coefficients of each group carefully, keeping the variable part unchanged.
第三步：仔细地对每一组的系数进行加减，变量部分保持不变。
Step 4: Write the final answer with the terms in a sensible order, usually with the highest power first.
第四步：按合理的顺序写出最终答案，通常把幂次最高的项写在最前面。
3a + 5b − a + 2b → (3a − a) + (5b + 2b) → 2a + 7b
Following the same routine every time reduces careless errors and saves time in the examination.
每次都遵循同样的流程，可以减少粗心错误，也能在考试中节省时间。
6. Working with Negative Terms | 处理负项
Most mistakes in this topic come from negative signs, not from the algebra itself. The rule is that the sign in front of a term travels with that term.
这一主题中的大多数错误来自负号，而不是代数本身。规则是：项前面的符号跟着这个项一起走。
re +8x and −3x, and the answer is 5x. In −6y + 2y, the answer is −4y, because −6 + 2 = −4.
因此在 8x − 3x 中，两项是 +8x 和 −3x，答案是 5x。在 −6y + 2y 中，答案是 −4y，因为 −6 + 2 = −4。
When a minus sign appears in front of a bracket, every sign inside the bracket changes, and only then can the terms be collected.
当括号前面出现减号时，括号内每一项的符号都要改变，之后才能合并同类项。
7x − (3x − 4) = 7x − 3x + 4 = 4x + 4
x = 1, will confirm that 7(1) − (3 − 4) = 8 and 4(1) + 4 = 8, so the answer is consistent.
用代入法快速检验，例如令 x = 1，可得 7(1) − (3 − 4) = 8，而 4(1) + 4 = 8，说明答案一致。
7. Powers, Different Letters and Products | 幂次、不同字母与乘积
Only terms with identical variable parts can be collected. x², x³ and x are three different families, and a, b and ab are three more, so each family must be handled separately.
只有变量部分完全相同的项才能合并。x²、x³ 和 x 是三个不同的"家族"，a、b 和 ab 又是另外三个，因此每个家族都要单独处理。
4x² + 3x − x² + 9x − 5 = 3x² + 12x − 5
he coefficients are multiplied and the powers are added, so 3x × 4x = 12x² rather than 7x.
乘法与加法完全不同。当项相乘时，系数相乘、幂次相加，因此 3x × 4x = 12x²，而不是 7x。
Division works in the opposite way: 6x² ÷ 2x = 3x, since 6 ÷ 2 = 3 and x² ÷ x = x.
除法则相反：6x² ÷ 2x = 3x，因为 6 ÷ 2 = 3，且 x² ÷ x = x。
This is why an expression such as 2x + 3x² + 5x − x² simplifies to 7x + 2x², and not to anything involving x³.
这就是为什么 2x + 3x² + 5x − x² 化简为 7x + 2x²，而不会出现任何 x³ 项。
8. Brackets: Expand First, Then Collect | 括号：先展开，再合并
If an expression contains brackets, expand the brackets first and collect like terms afterwards. Trying to collect through a bracket almost always leads to errors.
如果表达式含有括号，应先展开括号，然后再合并同类项。试图隔着括号直接合并几乎总会导致错误。
Expanding means multiplying the number outside the bracket by every term inside it, paying close attention to signs.
展开就是把括号外的数字乘以括号内的每一项，同时特别注意符号。
3(2x + 5) − 4(x − 1) = 6x + 15 − 4x + 4 = 2x + 19
ctangle of length 3x + 4 and width x + 1, the perimeter is 2(3x + 4) + 2(x + 1) = 8x + 10.
常见考题会给出用 x 表示的矩形边长。若矩形长为 3x + 4、宽为 x + 1，则周长为 2(3x + 4) + 2(x + 1) = 8x + 10。
Always write the expansion on a separate line before collecting, so that the marker can follow your reasoning and award method marks.
务必把展开过程单独写一行再进行合并，这样阅卷老师能看懂你的思路，给出方法分。
9. Why This Skill Matters | 为什么这项技能很重要
Collecting like terms is not an isolated skill. It appears in almost every later topic in IGCSE and A-Level Mathematics.
合并同类项不是一个孤立的技能，它几乎出现在 IGCSE 和 A-Level 数学之后的所有主题中。
efore isolating the unknown, for example 5x + 3 = 2x + 12 becomes 3x = 9 after collecting.
解方程时，你需要先在两边各自合并同类项，然后才把未知数分离出来，例如 5x + 3 = 2x + 12 合并后变为 3x = 9。
a messy expression into a clean equation such as y = 4x − 1, which is far easier to plot.
绘制图像时，合并同类项能把杂乱的表达式变成 y = 4x − 1 这样整洁的方程，画起来容易得多。
When rearranging formulas and answering word problems, you constantly need to recognise which quantities belong together.
在变形公式和解答应用题时，你需要不断判断哪些量属于同一类。
In short, this is a foundation skill that pays off in every later chapter.
总之，这是一项会在之后每个章节都带来回报的基础技能。
10. Common Mistakes to Avoid | 常见错误
Examiners report the same errors year after year. Knowing them in advance is the best protection.
考官年复一年地报告同样的错误。提前了解它们就是最好的防护。
- Adding unlike terms, for example writing 3x + 2 as 5x. | 把非同类项相加，例如把 3x + 2 写成 5x。
- Combining different powers, for example writing x² + x as x³. | 合并不同幂次，例如把 x² + x 写成 x³。
- Losing a minus sign when subtracting a bracket or a term. | 在减去括号或某一项时丢掉负号。
- Changing the variable part as well as the coefficient, for example 3x × 4x = 12x. | 既改了系数又改了变量部分，例如 3x × 4x = 12x。
- Forgetting that a term with no coefficient written has coefficient 1. | 忘记没有写系数的项其实系数是 1。
- Not writing the final answer in the simplest possible form. | 没有把最终答案写成最简形式。
as x = 2 into the original expression and into the answer, will catch most of these slips.
最后花三十秒做检验：把 x = 2 这样简单的值分别代入原表达式和答案，就能抓住上述大部分失误。
11. Worked Exam-Style Questions | 考试风格例题精讲
The following four questions cover the standard demand of IGCSE papers on this topic.
以下四道题覆盖了 IGCSE 试卷在这一主题上的标准要求。
up the a terms and the b terms: 5a − 2a = 3a and 3b + 7b = 10b, so the answer is 3a + 10b.
第 1 题：化简 5a + 3b − 2a + 7b。把 a 项和 b 项分别归类：5a − 2a = 3a，3b + 7b = 10b，所以答案是 3a + 10b。
Question 2: Simplify 6x² + 4x − x² + 9x − 5. The x² terms give 5x², the x terms give 13x, and the constant stays as −5, so the answer is 5x² + 13x − 5.
第 2 题：化简 6x² + 4x − x² + 9x − 5。x² 项得 5x²，x 项得 13x，常数项保持 −5，所以答案是 5x² + 13x − 5。
Question 3: Expand and simplify 3(2x + 5) − 4(x − 1). Expanding gives 6x + 15 − 4x + 4, and collecting gives 2x + 19.
第 3 题：展开并化简 3(2x + 5) − 4(x − 1)。展开得 6x + 15 − 4x + 4，合并后得 2x + 19。
6. Find the perimeter in its simplest form. Adding gives 2x + 3x + x + 1 − 4 + 6 = 6x + 3.
第 4 题：三角形的三边分别是 2x + 1、3x − 4 和 x + 6，求最简形式的周长。相加得 2x + 3x + x + 1 − 4 + 6 = 6x + 3。
12. Quick Revision Checklist | 快速复习清单
Before moving on to solving equations and inequalities, make sure you can tick every box below.
在进入解方程与不等式之前，请确保你能勾选下面每一项。
- I can break an expression into separate terms, keeping each sign. | 我能把表达式拆成单独的项，并保留每一项的符号。
- I can state the coefficient and the variable part of any term. | 我能说出任意一项的系数和变量部分。
- I can decide quickly whether two terms are like terms. | 我能迅速判断两项是否为同类项。
- I can collect like terms in expressions with several letters and powers. | 我能合并含多个字母和幂次的表达式中的同类项。
- I can expand brackets and then collect the result correctly. | 我能展开括号并正确合并结果。
- I can check my answer by substitution. | 我能用代入法检验答案。
Collecting like terms rewards accuracy rather than cleverness. Master the golden rule, keep your signs under control, and a great deal of later algebra becomes much easier.
合并同类项考验的是准确，而不是聪明。掌握黄金法则、管好符号，之后的大量代数内容都会变得轻松许多。
Like terms: same variable part → add or subtract the coefficients only
同类项：变量部分相同 → 只对系数做加减
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