📚 Collecting Like Terms in IGCSE Algebra | IGCSE 代数中的合并同类项
In IGCSE algebra, collecting like terms is the first essential skill for simplifying expressions. It means identifying terms that have exactly the same variables raised to the same powers, then adding or subtracting their coefficients. Every later topic - solving equations, expanding brackets, factorising, working with formulae - depends on doing this quickly and accurately. This guide takes you from the basic definition through to exam-level questions, including the traps that cost marks every session.
在 IGCSE 代数中，合并同类项是化简表达式的第一项基本技能。它是指找出含有相同变量且相同指数的项，然后将它们的系数相加或相减。后面几乎所有内容——解方程、去括号、因式分解、使用公式——都建立在这个技能之上，做得又快又准是前提。本文从最基本的定义讲起，一直讲到考试级别的题目，并特别指出每一届考试都会让学生丢分的陷阱。
1. What Are Like Terms? | 什么是同类项
A term is a single number, a single variable, or a product of numbers and variables. Examples of terms include 7, x, −3y, 4ab and 2x². Each term has two parts: a coefficient (the numerical factor, including its sign) and a variable part (the letters with their powers). In the term −3y the coefficient is −3 and the variable part is y¹; in 4ab the coefficient is 4 and the variable part is ab.
项是一个单独的数、一个单独的变量，或数与变量的乘积。项的例子包括 7、x、−3y、4ab 和 2x²。每个项由两部分组成：系数（数值因数，含正负号）和变量部分（字母及其指数）。在 −3y 中系数是 −3，变量部分是 y¹；在 4ab 中系数是 4，变量部分是 ab。
Like terms are terms that have exactly the same variable part: the same variables raised to the same powers. The coefficients can be different. For example, 3x and −5x are like terms because both contain x¹. Likewise 2a²b and −7a²b are like terms because both contain a²b with the letters in the same powers, even though the coefficients 2 and −7 are different.
同类项是变量部分完全相同的项：变量相同，且变量的指数也相同。系数可以不同。例如，3x 和 −5x 是同类项，因为两者都含有 x¹。同样，2a²b 和 −7a²b 也是同类项，因为两者都含有 a²b 且字母指数相同，尽管系数 2 与 −7 并不相同。
Unlike terms cannot be combined by addition or subtraction. For example, 4x and 4y are unlike terms because the variables are different. Similarly 6x and 6x² are unlike terms because the powers of x are different, and 4ab and 4a are unlike terms because the second term has no b. Adding unlike terms does not produce a simpler expression, so we simply leave them written side by side.
不同类项不能通过加法或减法合并。例如，4x 和 4y 是不同类项，因为变量不同。同样，6x 和 6x² 是不同类项，因为 x 的指数不同；4ab 和 4a 也是不同类项，因为第二项没有 b。把不同类项相加并不能得到更简单的表达式，所以只能把它们并列写出来。
2. Identifying Like Terms | 识别同类项
To identify like terms, ignore the coefficient and focus only on the variable factors and their powers. The variables must match exactly. Two useful questions to ask yourself are: which letters appear, and to what power is each letter raised? If both answers are identical, the terms are like terms.
识别同类项时，忽略系数，只关注变量因数及其指数。变量必须完全一致。可以问自己两个问题：出现了哪些字母？每个字母的指数是多少？如果两个答案完全一致，它们就是同类项。
| Terms | Like? | Reason |
|---|---|---|
| 4x and −9x | Yes | Both have x¹ |
| 3x² and 7x² | Yes | Both have x² |
| 5xy and 2yx | Yes | Multiplication order does not change the variables |
| 6x and 6x² | No | Different powers of x |
| 4ab and 4a | No | Different variable parts: ab versus a |
| −2x²y and 5x²y | Yes | Both have x²y |
| 3a²b and 3ab² | No | The powers are attached to different letters |
| 8 and −3 | Yes | Both are constants with no variable part |
Notice the last row: pure numbers, called constants, are always like terms with each other because their variable part is empty. Notice also the third row: 5xy and 2yx are like terms because xy and yx mean the same product. The order in which letters are written never changes what a term contains.
注意最后一行：纯数字称为常数项，它们的变量部分是空的，因此彼此之间总是同类项。再看第三行：5xy 与 2yx 是同类项，因为 xy 与 yx 表示同一个乘积。字母的书写顺序永远不会改变一个项所含的内容。
3. Why Collecting Like Terms Matters | 为什么要合并同类项
Collecting like terms turns a long, messy expression into the shortest possible equivalent expression. An answer such as 3x + 5y − x + 2y is correct but not finished; the simplified form 2x + 7y is what examiners expect and what makes later work easier.
合并同类项能把冗长杂乱的表达式变成与之等价的最短表达式。像 3x + 5y − x + 2y 这样的结果虽然正确，但没有完成；化简后的形式 2x + 7y 才是考官期待看到的答案，也让后续计算更轻松。
The skill is used constantly. When you solve a linear equation, you must simplify each side before isolating the unknown. When you expand brackets, you must collect the resulting terms. When you work with a formula in physics or mechanics, collecting like terms is what allows you to rearrange it into a usable form. In probability and statistics, simplifying algebraic expressions keeps formulae readable.
这项技能使用频率极高。解一次方程时，必须先化简两边，才能把未知数单独分离出来；去括号之后，必须合并得到的各项；在物理或力学的公式变形中，也正靠合并同类项才能把公式整理成可用的形式；在概率与统计里，化简代数表达式能让公式保持易读。
Finally, collecting like terms is the foundation of the topic of algebraic manipulation at A-Level. Adding and subtracting algebraic fractions, completing the square, and simplifying polynomial expressions all rely on being able to group terms correctly by their variable part. Students who are fluent here find later algebra far less intimidating.
最后，合并同类项是 A-Level 代数运算专题的基础。分式加减、配方法、多项式化简，都依赖于能否正确地按变量部分把项分组。在这一步熟练的学生，面对后面的代数内容会觉得轻松得多。
4. The Coefficient Rule | 系数法则
ged. The letters are not touched at all. This works because of the distributive law, since a xⁿ + b xⁿ = (a + b) xⁿ.
整个方法建立在一个简单法则上：合并同类项时，把系数相加或相减，变量部分保持不变。字母本身完全不动。这样做之所以成立，是因为乘法分配律，即 a xⁿ + b xⁿ = (a + b) xⁿ。
a xⁿ + b xⁿ = (a + b) xⁿ and a xⁿ − b xⁿ = (a − b) xⁿ
and someone gives you 5 apples, you have 8 apples: 3 apples + 5 apples = 8 apples. Replace "apples" ith x and you get 3x + 5x = 8x. The object never changes - only how many of them you have.
一个很实用的想象方式是数东西。如果你有 3 个苹果，别人又给你 5 个苹果，你就有了 8 个苹果：3 个苹果 + 5 个苹果 = 8 个苹果。把"苹果"换成 x，就得到 3x + 5x = 8x。东西本身从不改变，改变的只是数量。
This analogy also explains why x + x² is not 2x². One x and one x² are different objects, just as one apple and one orange cannot be added into two of either fruit. In the same way, 3ab + 4a cannot be simplified, because the objects counted are different.
这个类比也解释了为什么 x + x² 不等于 2x²。一个 x 和一个 x² 是两种不同的东西，就像一个苹果和一个橘子不能加成一个"两倍的东西"。同理，3ab + 4a 也无法化简，因为所计数的对象并不相同。
5. Step-by-Step Method | 分步方法
Follow these four steps every time, and the process becomes automatic. Step 1: rewrite the expression treating every subtraction as adding a negative, so that each term is clearly signed. Step 2: group the terms that share the same variable part, using underlining, circling or colour to keep track.
每次都按下面四步走，这个过程就会变成自动反应。第一步：把每个减法都看作加上一个负数，重写整个表达式，让每个项的正负号都清楚明确。第二步：把变量部分相同的项分组，可以用下划线、圈选或不同颜色来跟踪。
Step 3: within each group, add or subtract the coefficients, keeping the variable part unchanged. Step 4: write the simplified result in a standard order, normally with the highest powers first and the constant term last, and check that no term has been lost.
第三步：在每一组内部把系数相加或相减，变量部分保持不变。第四步：把化简结果按标准顺序写出，通常指数高的项在前、常数项在最后，并检查有没有漏掉任何一项。
- Rewrite with signs attached / 重写时让正负号跟着项走
- Group terms with identical variable parts / 把变量部分相同的项分组
- Combine coefficients only / 只合并系数
- Order the answer and check the number of terms / 整理顺序并检查项数
d after. Terms may only disappear if their coefficients cancel to zero, as in 5x − 5x = 0.
一个快速检查方法：比较化简前后的项数。只有系数相消为零时，项才会消失，例如 5x − 5x = 0。
6. Worked Examples: Linear Expressions | 例题：一次表达式
+ 8b. The two results cannot be combined further because a and b are different variables.
例 1：化简 7a + 3b − 2a + 5b。把含 a 的项归为一组：7a − 2a = 5a。把含 b 的项归为一组：3b + 5b = 8b。答案是 5a + 8b。由于 a 与 b 是不同变量，这两个结果无法再合并。
is 3x − 3. Writing 3x + (−3) is also acceptable, but 3x − 3 is the standard finished form.
例 2：化简 4x + 9 − x − 12。含 x 的项得到 4x − x = 3x。常数项得到 9 − 12 = −3。答案是 3x − 3。写成 3x + (−3) 也可以，但 3x − 3 是标准答案形式。
mpletely. Only the constant remains, so the answer is 2. Example 4: simplify 5m + 3n − 2m − 8n + 4. The m terms give 3m, the n terms give −5n, and the constant stays as 4, so the answer is 3m − 5n + 4.
例 3：化简 −6y + 2 + 6y。含 y 的项得到 −6y + 6y = 0，完全相消。只剩下常数项，所以答案是 2。例 4：化简 5m + 3n − 2m − 8n + 4。含 m 的项得 3m，含 n 的项得 −5n，常数项仍为 4，所以答案是 3m − 5n + 4。
7. Handling Negative Signs and Brackets | 处理负号与括号
Most lost marks in this topic come from sign errors, not from misunderstanding the concept. When a minus sign sits in front of a bracket, every term inside the bracket changes sign when the bracket is removed. For example, 5x − (3x − 4y) + 2y becomes 5x − 3x + 4y + 2y, which simplifies to 2x + 6y.
本专题中大部分失分来自符号错误，而不是概念不清。当减号出现在括号前面时，去掉括号后括号内每一项都要变号。例如 5x − (3x − 4y) + 2y 变为 5x − 3x + 4y + 2y，化简得 2x + 6y。
When a number multiplies a bracket, multiply every term inside before collecting. Example: 2(3a − b) − 4(a − 2b). Expanding gives 6a − 2b − 4a + 8b. Collecting the a terms gives 2a, and collecting the b terms gives 6b, so the answer is 2a + 6b.
当括号前有数字相乘时，要先把括号内每一项都乘开，再合并。例如：2(3a − b) − 4(a − 2b)。展开得 6a − 2b − 4a + 8b。合并含 a 的项得 2a，合并含 b 的项得 6b，所以答案是 2a + 6b。
Pay special attention to the second bracket above. Multiplying by −4 changes the sign of both a and −2b, giving −4a and +8b. A frequent error is to write −8b instead of +8b. Writing out the intermediate step costs a few seconds but saves the mark.
要特别注意上面第二个括号。乘以 −4 会同时改变 a 和 −2b 的符号，得到 −4a 和 +8b。常见错误是把 +8b 写成 −8b。多写一步中间过程只花几秒钟，却能保住分数。
8. Like Terms with Powers and Multiple Variables | 含指数与多变量的同类项
cted, because x² and x are different variable parts. Remember that x means x¹, so x and x² are never like terms.
含指数的项遵循完全相同的法则，但指数必须完全一致。所以 3x² + 7x² = 10x²，9p³ − 4p³ = 5p³。然而 3x² + 7x 无法合并，因为 x² 与 x 是不同的变量部分。记住 x 就是 x¹，所以 x 和 x² 永远不是同类项。
s multiply and the powers add, giving x⁴. These two operations must never be mixed up. A useful habit is to check whether the terms are separated by a plus or minus sign (addition, collect like terms) or by a multiplication sign (indices rule).
不要把指数相加与指数相乘弄混。在 x² + x² 中指数不变、系数相加，得到 2x²。在 x² × x² 中系数相乘、指数相加，得到 x⁴。这两种运算绝不可混用。一个好习惯是先看各项之间是加减号（属于合并同类项）还是乘号（属于指数法则）。
t 3x²y and 3xy² are not like terms, because in the first term x is squared and in the second term y is squared. Here is a mixed example: 3x² + 5x − 2x² + 7x − 4 simplifies to x² + 12x − 4.
对于含有多个变量的项，每个字母及其指数都必须一致。例如：3x²y + 5x²y − 2x²y = 6x²y。但 3x²y 与 3xy² 不是同类项，因为前一个式子中 x 是平方，后一个式子中 y 是平方。再看一个混合例子：3x² + 5x − 2x² + 7x − 4 化简为 x² + 12x − 4。
9. Common Mistakes and Exam Traps | 常见错误与考试陷阱
part. Writing 3x + 5x = 8x² is wrong; the answer is 8x, because only the coefficients add.
错误一：把不同类项合并。写成 x + x² = 2x² 是错的，正确答案就是 x + x²，保持原样。错误二：改变变量部分。写成 3x + 5x = 8x² 是错的，正确答案是 8x，因为只有系数相加。
Error 3: dropping a negative sign when a bracket is removed. Error 4: forgetting to multiply every term inside a bracket by the factor outside. Error 5: leaving the answer unsimplified when the question asks for the simplest form, which often loses the final mark even when the arithmetic is right.
错误三：去括号时漏掉负号。错误四：忘记用括号外的因数乘括号内的每一项。错误五：题目要求最简形式，却没有化简到底，即使计算正确也常常丢掉最后一分。
not 7xy. If you spot a multiplication sign anywhere between two terms, you must not collect them by adding coefficients. Error 7: writing answers in a non-standard order; while 7y + 2x is mathematically correct, examiners usually expect descending powers, so 2x + 7y is the safer form.
错误六：把乘积看成和。像 4x × 3y 这样的表达式等于 12xy，而不是 7xy。只要两个项之间出现乘号，就绝不能通过相加系数来合并。错误七：答案顺序不规范；虽然 7y + 2x 在数学上正确，但考官通常期待降幂排列，所以 2x + 7y 更稳妥。
10. Exam-Style Questions with Solutions | 考试风格题目与解答
ion 2: Simplify 6x² − 4x + 3 − 2x² + 9x − 7. Solution: x² terms give 4x², x terms give 5x, constants give −4, so the answer is 4x² + 5x − 4.
题 1：化简 8a − 3b + 2a − 5b。解答：含 a 的项合并得 8a + 2a = 10a，含 b 的项合并得 −3b − 5b = −8b。答案是 10a − 8b。题 2：化简 6x² − 4x + 3 − 2x² + 9x − 7。解答：x² 项得 4x²，x 项得 5x，常数项得 −4，所以答案是 4x² + 5x − 4。
n by 2(l + w). If l = 3x + 1 and w = x − 2, find the perimeter in simplest form. Solution: l + w = 4x − 1, so the perimeter is 2(4x − 1) = 8x − 2.
题 3：化简 3(2x − 5) − 2(x − 4)。解答：展开得 6x − 15 − 2x + 8。合并得 4x − 7。题 4：矩形周长为 2(l + w)。若 l = 3x + 1，w = x − 2，求周长的最简形式。解答：l + w = 4x − 1，所以周长为 2(4x − 1) = 8x − 2。
x = 0, and the y terms give 10y − 4y − 6y = 0. Hence the expression equals 0, as required.
题 5：证明 5(x + 2y) − 4(x + y) − (x + 6y) = 0。解答：展开得 5x + 10y − 4x − 4y − x − 6y。合并含 x 的项得 5x − 4x − x = 0，含 y 的项得 10y − 4y − 6y = 0。因此该表达式等于 0，得证。
11. Practice Set | 练习题
Work through the following ten questions without a calculator. They cover everything discussed above: basic collection, constants, powers, multiple variables, brackets and negative signs. Aim for accuracy first, then speed.
请在不使用计算器的情况下完成下面十道题。它们覆盖了上文讨论的全部内容：基本合并、常数项、指数、多变量、括号与负号。先追求正确，再追求速度。
- 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
