📚 Collecting Like Terms: Complete Guide and Exam Practice | 合并同类项：完整指南与考试练习
Collecting like terms is one of the very first algebraic skills taught at IGCSE, yet it reappears in almost every later topic: solving equations, expanding brackets, factorising, rearranging formulae, simplifying algebraic fractions, and differentiating in A-Level. This guide explains how to identify like terms, how to combine them correctly, and how to avoid the small errors that quietly cost marks in examinations. Every section is written for both IGCSE and A-Level students who want a reliable, exam-ready method that works every time.
合并同类项是 IGCSE 阶段最早接触的代数技能之一，但它几乎出现在之后所有知识点中：解方程、去括号、因式分解、公式变形、代数分式化简，以及 A-Level 求导。本文将讲解如何识别同类项、如何正确合并，以及如何避免那些在考试中悄悄丢分的小错误。每一节都同时适合 IGCSE 与 A-Level 学生，帮助你建立可靠、可直接用于考试的方法。
1. Terms, Coefficients and Variables | 项、系数与变量
An algebraic expression is built from terms. A term can be a single number, a single letter, or a product of numbers and letters. Examples include 5x, −3y, 7x², 4xy and 9. The number in front of the letters is called the coefficient. In 5x the coefficient is 5; in −3y the coefficient is −3. Always treat the sign as part of the coefficient, never as something separate.
代数表达式由项组成。项可以是一个数字、一个字母，也可以是数字和字母的乘积。例如 5x、−3y、7x²、4xy 和 9。字母前面的数字称为系数。在 5x 中系数是 5；在 −3y 中系数是 −3。符号始终要算作系数的一部分，绝不能把它和系数分开看待。
Variables are the unknown letters such as x, y, a, b or t. A constant term is a term with no variable, such as 4 or −7. Being able to split every term into its coefficient, its variable part and its exponents makes collecting like terms far easier, because you only ever add the coefficients and never touch the letters.
变量是表示未知数的字母，例如 x、y、a、b 或 t。常数项是没有字母的项，例如 4 或 −7。能够把每一项拆分为系数、字母部分和指数，会让合并同类项变得容易得多，因为你永远只需要处理系数，字母部分保持不变。
| Term | Coefficient | Variable part | Exponent(s) |
|---|---|---|---|
| 6x² | 6 | x | 2 |
| −4ab | −4 | ab | 1 and 1 |
| x | 1 | x | 1 |
| −7 | −7 | none (constant) | — |
Notice the third row: the term x has coefficient 1, not 0. Writing 1x is unnecessary but remembering the hidden 1 prevents a very common counting error. For the term 6x², the coefficient is 6, the variable is x and the exponent is 2. For the term −4ab, the coefficient is −4 and the variable part is ab.
注意表格第三行：项 x 的系数是 1，而不是 0。写成 1x 没有必要，但记住这个隐藏的 1，可以避免一个非常常见的计数错误。对于 6x² 这一项，系数是 6，变量是 x，指数是 2。对于 −4ab 这一项，系数是 −4，字母部分是 ab。
2. What Makes Terms 'Like'? | 什么是“同类项”？
Two or more terms are called like terms when they have exactly the same variable part and each variable has the same power. The coefficients can be completely different. For example, 3x and 7x are like terms because both contain x with power 1, and 5a²b and −2a²b are like terms because both contain a²b.
两个或多个项，如果字母部分完全相同且每个字母的指数也相同，就称为同类项。系数可以完全不同。例如 3x 和 7x 是同类项，因为二者都含有 x 且指数为 1；5a²b 和 −2a²b 也是同类项，因为二者都含有 a²b。
Unlike terms have different variable parts or different powers. For instance, 3x and 3y are not like terms because the variables differ, and 2x and 2x² are not like terms because the powers of x differ. The test is always about the letters and their exponents, never about the numbers in front.
不同类项的字母部分或指数不同。例如 3x 和 3y 不是同类项，因为变量不同；2x 和 2x² 也不是同类项，因为 x 的指数不同。判断标准始终是字母及其指数，而与前面的数字无关。
| Terms | Like or not? | Reason |
|---|---|---|
| 3x and 7x | Like | Same variable x, same power 1 |
| 3x and 3y | Not like | Different variables |
| 2x and 2x² | Not like | Different powers of x |
| 5ab and 3ba | Like | ab = ba, so the variable part is identical |
| 4x²y and 4xy² | Not like | In the first, x has power 2; in the second, y does |
| 6 and −9 | Like | Both are constants with the same (empty) variable part |
| x and 1 | Not like | One is a variable term, the other a constant |
The fourth row is worth memorising: multiplication is commutative, so ab and ba are the same product. Many students lose marks simply because they refuse to see 3ba and 5ab as matching terms.
表格第四行值得牢记：乘法满足交换律，所以 ab 和 ba 是同一个乘积。很多学生丢分，仅仅是因为没有意识到 3ba 和 5ab 其实是同类项。
3. The Core Rule: Add the Coefficients, Keep the Letters | 核心法则：系数相加，字母不变
The whole technique rests on one idea, which comes directly from the distributive law. If two terms have the same variable part, you can factor that part out and add the numbers in front.
整套方法建立在一个核心思想之上，它直接来自乘法分配律。如果两项的字母部分相同，就可以把这个字母部分提取出来，只把前面的数字相加。
ax + bx = (a + b)x
3x + 7x = (3 + 7)x = 10x
Read that second line as a sentence: three x's plus seven x's makes ten x' because subtracting is the same as adding a negative coefficient: 9y − 4y = (9 − 4)y = 5y.
把第二行读成一句话：3 个 x 加 7 个 x 等于 10 个 x。减法也遵循同样的逻辑，因为减去一个数等于加上它的相反数：9y − 4y = (9 − 4)y = 5y。
- If the coefficients add to zero, the terms cancel completely: 5k − 5k = 0.
- If the result is negative, write the negative sign clearly: 3m − 8m = −5m.
- Never add the exponents when collecting: x² + x² = 2x², not x⁴.
- Never merge different letters: 3x + 4y stays as 3x + 4y.
- 如果系数相加得零，这些项完全抵消：5k − 5k = 0。
- 如果结果是负数，要把负号写清楚：3m − 8m = −5m。
- 合并同类项时绝不要把指数相加：x² + x² = 2x²，而不是 x⁴。
- 绝不合并不同的字母：3x + 4y 就保持为 3x + 4y。
4. Worked Examples with a Single Variable | 单变量例题精讲
Start with expressions containing only one letter. The method is always the same: scan the expression, group the terms with identical variable parts, add the coefficients of each group, then write the simplified expression with the highest power first (descending order).
先从只含一个字母的表达式开始。方法始终相同：先扫视整个表达式，把字母部分完全相同的项归为一组，把每组系数相加，最后按指数从高到低写出化简结果。
Example 1: Simplify 5x + 3x − 2x.
例 1：化简 5x + 3x − 2x。
terms are like terms, so add the coefficients: 5 + 3 − 2 = 6. Therefore 5x + 3x − 2x = 6x.
三项都是同类项，所以把系数相加：5 + 3 − 2 = 6。因此 5x + 3x − 2x = 6x。
Example 2: Simplify 8a − 3 + 2a + 7.
例 2：化简 8a − 3 + 2a + 7。
wer is 10a + 4. The two groups cannot be merged because a and a number are not like terms.
把含 a 的项分为一组：8a + 2a = 10a。把常数分为一组：−3 + 7 = 4。答案是 10a + 4。这两组不能合并，因为字母项和常数项不是同类项。
Example 3: Simplify 4x² + 3x + 2x² − 7x.
例 3：化简 4x² + 3x + 2x² − 7x。
simplified expression is 6x² − 4x. Answer written as 6x² − 4x, with the square term first.
按指数分组：4x² + 2x² = 6x²，3x − 7x = −4x。化简结果为 6x² − 4x。书写时把二次项写在前面，即 6x² − 4x。
5. Collecting Like Terms with Multiple Variables | 含多个字母的合并同类项
When an expression contains several letters, you may still collect, but only within identical variable parts. Each distinct variable part forms its own group, and the groups stay separate in the final answer.
当表达式含有多个字母时，仍然可以合并，但只能在字母部分完全相同的项之间进行。每一种不同的字母部分各成一组，各组在最终答案中保持分开。
Example 4: Simplify 3x + 4y + 5x − 2y.
例 4：化简 3x + 4y + 5x − 2y。
riting 10xy would be wrong, because x and y are different variables and their product has a different meaning altogether.
含 x 的项得到 3x + 5x = 8x。含 y 的项得到 4y − 2y = 2y。答案是 8x + 2y。写成 10xy 是错误的，因为 x 与 y 是不同变量，它们的乘积含义完全不同。
Example 5: Simplify 7ab − 2a + 3ba + 6a.
例 5：化简 7ab − 2a + 3ba + 6a。
answer is 10ab + 4a. Note that ab and a are not like terms, because the variable parts differ.
因为 ba = ab，所以含 ab 的项为 7ab + 3ab = 10ab。含 a 的项为 −2a + 6a = 4a。答案是 10ab + 4a。注意 ab 和 a 不是同类项，因为字母部分不同。
Example 6: Simplify 5x²y + 2xy² − x²y + 4xy².
例 6：化简 5x²y + 2xy² − x²y + 4xy²。
is 4x²y + 6xy², and the two groups must remain separate because the exponents sit on different letters.
含 x²y 的项：5x²y − x²y = 4x²y。含 xy² 的项：2xy² + 4xy² = 6xy²。答案是 4x²y + 6xy²。这两组必须保持分开，因为指数落在不同的字母上。
6. Powers and Exponents: Why x and x² Never Merge | 幂与指数：为什么 x 与 x² 不能合并
Exponents cause more lost marks than any other part of this topic. The rule is simple and worth stating explicitly: only terms with exactly the same exponent on every variable can be collected. x, x², x³ and x⁴ are four different variable parts, so they form four separate groups.
指数是本知识点中造成丢分最多的部分。规则很简单，值得明确说明：只有当每个字母的指数都完全相同时，才能合并。x、x²、x³、x⁴ 是四种不同的字母部分，因此构成四组，彼此不能合并。
x² + x² = 2x²
x² + x³ cannot be simplified
t when you add powers you only add the coefficients. Confusing these two rules is the single most common error in early algebra.
合并同类项属于加法，而不是乘法。同底数幂相乘时指数相加，例如 x² × x³ = x⁵；但同底数幂相加时，只把系数相加。混淆这两条规则，是初学代数中最常见的错误。
| Expression | Correct result | Common wrong answer |
|---|---|---|
| 3x² + 4x² | 7x² | 7x⁴ |
| 2x + 5x² | 2x + 5x² (unchanged) | 7x³ |
| 4x² × 3x² | 12x⁴ | 12x² |
| 6x³ − x³ | 5x³ | 6 |
7. Handling Negative Signs and Subtraction | 处理负号与减法
Negative coefficients are the second biggest source of errors. The safest habit is to keep the sign attached to the term and mentally rewrite every subtraction as an addition of a negative number before you begin collecting.
负系数是第二大错误来源。最稳妥的习惯是让符号始终跟随该项，并在开始合并之前，在脑中把每一个减法都改写成加上一个负数。
Example 7: Simplify 6x − 8x + 3x.
例 7：化简 6x − 8x + 3x。
(which means 1x). If you had mistakenly computed 6 − 8 + 3 as 6 + 8 + 3, you would have written 17x, a completely different expression.
改写成 6x + (−8x) + 3x。系数计算为 6 − 8 + 3 = 1，所以答案是 x（即 1x）。如果把 6 − 8 + 3 误算成 6 + 8 + 3，就会写出 17x，那将是一个完全不同的表达式。
Example 8: Simplify −4x + 9x − 12x.
例 8：化简 −4x + 9x − 12x。
the answer is −7x. A useful check is to estimate the size of the result: two negative contributions and one positive contribution should leave a negative total, which matches −7.
系数为 −4 + 9 − 12 = −7，所以答案是 −7x。一个有用的检验方法是估算结果的正负：两个负数贡献加一个正数贡献，结果应为负，这与 −7 相符。
- Write the sign in front of the coefficient: −7x, not 7x−.
- If the coefficient is 1 or −1, still write the sign: the answer is −x, not x.
- When a term has no visible sign, it is positive: 5y means +5y.
- Check by substituting a number, for example x = 2, into both versions.
- 把符号写在系数前面：写作 −7x，而不是 7x−。
- 系数为 1 或 −1 时，符号仍要保留：答案是 −x，而不是 x。
- 当某项没有可见符号时，它是正项：5y 意为 +5y。
- 可以代入数字检验，例如把 x = 2 代入原式和化简式比较。
8. Simplifying Expressions with Brackets | 带括号表达式的化简
Many exam questions combine expansion with collecting. The correct order is always: expand every bracket first, then collect like terms, then write the answer in descending order. Never try to collect terms across a bracket before it has been removed.
许多考题把去括号与合并同类项结合起来。正确的顺序永远是：先去括号，再合并同类项，最后按指数降序写出答案。在没有去掉括号之前，绝不能跨括号合并。
Example 9: Simplify 2(3x + 4) + 3(x − 2).
例 9：化简 2(3x + 4) + 3(x − 2)。
8, and 3(x − 2) = 3x − 6. Now collect: 6x + 3x = 9x, and 8 − 6 = 2. The answer is 9x + 2.
去括号：2(3x + 4) = 6x + 8，3(x − 2) = 3x − 6。然后合并：6x + 3x = 9x，8 −
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