Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, the expression we see is rarely in its simplest form. Often, multiple terms can be combined into one shorter, clearer expression. This process is called collecting like terms or simplifying algebraic expressions. It is one of the first and most essential skills in IGCSE mathematics, as it appears in almost every algebraic topic that follows.

在代数中，我们看到的表达式很少是最简形式。通常，多个项可以合并成一个更短、更清晰的表达式。这个过程称为合并同类项或化简代数表达式。这是IGCSE数学中最基础也最重要的技能之一，因为后续几乎所有代数主题中都会用到它。

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. The coefficients — the numbers in front of the variables — can be different. For example, 3x and 5x are like terms because both have the variable x raised to the first power. Similarly, 2x² and -7x² are like terms because both contain x squared. The numbers 4 and -9 are also like terms, because both are constant terms with no variable attached.

同类项是指变量部分完全相同的项，包括相同的幂次。系数——变量前的数字——可以不同。例如，3x 和 5x 是同类项，因为两者都含有一次的变量 x。同样，2x² 和 -7x² 是同类项，因为两者都含有 x 的平方。数字 4 和 -9 也是同类项，因为两者都是不带变量的常数项。

Terms that are not like terms cannot be combined. For instance, x and x² are not like terms, because the powers of x are different. The variable y is completely different from x, so x and y cannot be combined either. A constant term like 5 cannot be combined with 3x, because 5 has no variable part at all.

不是同类项的项不能合并。例如，x 和 x² 不是同类项，因为 x 的幂次不同。变量 y 与 x 完全不同，所以 x 和 y 也不能合并。常数项如 5 不能与 3x 合并，因为 5 根本没有变量部分。

2. Why Do We Collect Like Terms? | 为什么要合并同类项？

There are several important reasons to collect like terms. First, a simplified expression is easier to read and understand. Take the expression 4x + 3 + 2x – 1. As written, it requires two separate mental operations. By combining the x terms and the constant terms, we get 6x + 2, which is much clearer and shorter. Second, simplification helps us avoid careless arithmetic errors when we later substitute values or manipulate equations. Finally, in solving equations, simplification often reveals the structure of the problem directly.

合并同类项有几个重要原因。第一，化简后的表达式更容易阅读和理解。以表达式 4x + 3 + 2x – 1 为例，按原样需要两步心算。通过合并 x 项和常数项，我们得到 6x + 2，更加清晰简洁。第二，化简有助于避免后续代入数值或处理方程时出现粗心算术错误。最后，在解方程时，化简往往能直接揭示问题的结构。

3. The Method | 合并的方法

The method for collecting like terms is simple and systematic. First, identify all the like terms by looking at the variable part of each term. Second, add or subtract the coefficients of the like terms. Third, keep the variable part unchanged. Finally, write the simplified expression with terms in descending order of power, usually with the constant term last.

合并同类项的方法简单而系统。首先，通过观察每个项的变量部分来识别所有同类项。其次，对同类项的系数进行加减运算。第三，保持变量部分不变。最后，按幂次降序书写化简后的表达式，通常把常数项放在最后。

4x + 7x = (4 + 7)x = 11x

5x² – 2x² = (5 – 2)x² = 3x²

3x + 4 + 2x + 5 = 5x + 9

Notice that in the third example, we combine the x terms separately from the constant terms. The variables never change when we combine terms. We are only adding or subtracting the coefficients — the multiplier in front of each variable.

注意在第三个例子中，我们分别合并 x 项和常数项。合并项时变量从不改变。我们只对系数——每个变量前的乘数——进行加减运算。

4. Worked Examples | 工作示例

Let us look at several worked examples to see the method in action.

让我们看几个完整的示例，了解该方法的实际运用。

Example 1: Simplify 6a + 3b – 2a + b.

示例 1：化简 6a + 3b – 2a + b。

Identify the like terms: 6a and -2a are like terms; 3b and +b are like terms. Now combine each group. For the a terms: 6 – 2 = 4, giving 4a. For the b terms: 3 + 1 = 4, giving 4b. Notice that b means 1b. The simplified expression is 4a + 4b.

识别同类项：6a 和 -2a 是同类项；3b 和 +b 是同类项。然后分别合并：对于 a 项，6 – 2 = 4，得到 4a；对于 b 项，3 + 1 = 4，得到 4b。注意 b 即 1b。化简后的表达式为 4a + 4b。

Example 2: Simplify 5x² + 3x – 2x² + 7.

示例 2：化简 5x² + 3x – 2x² + 7。

Here, 5x² and -2x² are like terms, giving 3x². There is only one x term, which is 3x. The constant is 7. So the answer is 3x² + 3x + 7. Note that x² terms and x terms are not like terms, so they remain separate.

这里，5x² 和 -2x² 是同类项，合并得 3x²。只有一个 x 项，即 3x。常数项是 7。所以答案是 3x² + 3x + 7。注意 x² 项和 x 项不是同类项，因此它们保持分离。

Example 3: Simplify 4(2x + 3) – 2(3x – 1).

示例 3：化简 4(2x + 3) – 2(3x – 1)。

This example involves brackets. We must first expand the brackets before collecting like terms. Expanding 4(2x + 3) gives 8x + 12. Expanding -2(3x – 1) gives -6x + 2. Be careful with the negative sign: -2 multiplied by -1 is +2. Now combine 8x and -6x to get 2x, and combine 12 and 2 to get 14. The answer is 2x + 14.

这个示例涉及括号。我们必须先展开括号，然后才能合并同类项。展开 4(2x + 3) 得 8x + 12；展开 -2(3x – 1) 得 -6x + 2。注意负号：-2 乘以 -1 得 +2。然后合并 8x 和 -6x 得 2x，合并 12 和 2 得 14。答案为 2x + 14。

5. Common Mistakes to Avoid | 常见错误与陷阱

Even strong students make errors when collecting like terms. Here are the most common mistakes and how to avoid them.

即使是优秀的学生在合并同类项时也会犯错。以下是最常见的错误及避免方法。

Mistake 1: Combining x and x². These are not like terms. x means 1x¹ and x² means 1x². The powers are different. Always check the exponent of each variable. If the exponent differs, the terms cannot be combined.
错误 1：合并 x 和 x²。它们不是同类项。x 表示 1x¹，x² 表示 1x²。幂次不同。始终检查每个变量的指数。若指数不同，则不能合并。
Mistake 2: Forgetting the invisible 1. A term like y has a coefficient of 1. When adding y + y, do not write y². The correct sum is 2y. The coefficient 1 is not usually written, but it is always there.
错误 2：忘记隐形的系数 1。像 y 这样的项系数为 1。当 y + y 相加时，不要写成 y²。正确结果是 2y。系数 1 通常不写出来，但它始终存在。
Mistake 3: Sign errors with negative terms. In an expression like 3x – 5x, many students write -8x instead of -2x. Remember: 3 – 5 = -2. The negative sign belongs to the coefficient, not the whole term. Treat every term as having a positive or negative coefficient, and add those signed coefficients.
错误 3：负号运算错误。在 3x – 5x 这样的表达式中，许多学生写 -8x 而不是 -2x。记住：3 – 5 = -2。负号属于整个系数。把每一项视为带有正号或负号的系数，再对这些带符号的系数进行相加。
Mistake 4: Changing the variable part. When adding 2x and 3x, the result is 5x, not 5x². The variable and its exponent never change. Only the coefficient changes.
错误 4：改变变量部分。2x 加 3x 的结果是 5x，不是 5x²。变量及其指数永远不变。只有系数改变。

6. Collecting Like Terms with Higher Powers | 含高次幂的合并

In IGCSE, expressions can contain powers higher than 2, such as x³ or x⁴. The same rules apply: terms are like terms only when the variable and its exponent match exactly. For example, 4x³ and -2x³ are like terms and combine to give 2x³. However, x³ and x² cannot be combined, nor can x³ and x.

在IGCSE中，表达式可能包含高于2的幂，例如 x³ 或 x⁴。同样的规则适用：只有当变量及其指数完全匹配时，项才是同类项。例如，4x³ 和 -2x³ 是同类项，合并得 2x³。然而，x³ 和 x² 不能合并，x³ 和 x 也不能合并。

Consider the expression 2x³ + 4x² – x³ + 3x. First, combine 2x³ and -x³ to get x³. The x² term and the x term each stand alone. The simplified expression is x³ + 4x² + 3x. Notice we order terms from highest power to lowest power for a polished final answer.

考虑表达式 2x³ + 4x² – x³ + 3x。首先合并 2x³ 和 -x³ 得 x³。x² 项和 x 项各自独立。化简后的表达式为 x³ + 4x² + 3x。注意我们按从高到低的幂次排列各项，使最终答案更规范。

A useful strategy is to identify terms with the same variable and same power, group them mentally, and then combine. You might find it helpful to underline terms of the same type in different colours when working on paper.

一个有用的策略是：识别具有相同变量和相同幂次的项，在脑中分组，然后合并。在纸上做题时，可以用不同颜色下划线标记同类型的项，这会很有帮助。

7. The Distributive Law and Collecting | 分配律与合并的结合

The distributive law states that a(b + c) = ab + ac. When an expression contains brackets, we must first apply the distributive law to remove the brackets, and only then can we collect like terms. Failing to expand brackets first is a common error.

分配律指出 a(b + c) = ab + ac。当表达式中含有括号时，我们必须先运用分配律去掉括号，然后才能合并同类项。忘记先展开括号是一个常见错误。

Example: Simplify 3(2x + 1) – 2(x – 4).

示例：化简 3(2x + 1) – 2(x – 4)。

Step 1: Expand the brackets. 3(2x + 1) = 6x + 3. And -2(x – 4) = -2x + 8. Step 2: Combine like terms. 6x – 2x = 4x and 3 + 8 = 11. The simplified expression is 4x + 11.

第一步：展开括号。3(2x + 1) = 6x + 3，且 -2(x – 4) = -2x + 8。第二步：合并同类项。6x – 2x = 4x，3 + 8 = 11。化简后的表达式为 4x + 11。

When the coefficient outside the brackets is negative, be extra careful with the signs inside. Multiplying -2 by -4 gives +8, not -8. Many sign errors occur at this exact step.

当括号外的系数为负时，要格外小心括号内的符号。注意 -2 乘以 -4 得 +8，而不是 -8。许多符号错误正是在这一步发生。

8. Applications in Solving Equations | 在解方程中的应用

Collecting like terms is essential in solving linear equations. For example, consider 3x + 5 = 2x + 9. The goal is to isolate x. First, subtract 2x from both sides to collect the x terms on the left: x + 5 = 9. Then subtract 5 from both sides: x = 4. Without the ability to simplify expressions, this process would be much more difficult.

合并同类项在解线性方程中至关重要。例如，考虑 3x + 5 = 2x + 9。目标是隔离 x。首先，两边同时减去 2x，使 x 项在左边合并：x + 5 = 9。然后两边同时减去 5：x = 4。没有化简表达式的能力，这个过程会困难得多。

In equations with variables on both sides, the first step is always to simplify each side by collecting like terms, then use inverse operations to move all variable terms to one side and all constant terms to the other.

在变量同时出现在等号两边的方程中，第一步始终是分别化简等号两边的表达式，然后使用逆运算将所有变量项移到一边，所有常数项移到另一边。

Consider a slightly harder equation: 4(2x + 3) – 2(3x – 1) = 30. Using what we did in Example 3, simplify the left side to 2x + 14. Now solve 2x + 14 = 30. Subtract 14 and divide by 2, giving x = 8. Simplifying first made the equation manageable.

考虑一个稍难的方程：4(2x + 3) – 2(3x – 1) = 30。利用示例 3 中的方法，将左边化简为 2x + 14。现在解 2x + 14 = 30。两边减去 14 再除以 2，得到 x = 8。先化简使方程更加易解。

9. Real-World Contexts | 实际应用场景

Algebraic simplification appears in many practical situations. Suppose a rectangle has length 3x + 2 and width x + 1. Its perimeter is 2(length + width) = 2[(3x + 2) + (x + 1)] = 2(4x + 3) = 8x + 6. Here, collecting like terms inside the brackets simplified the calculation.

代数化简出现在许多实际场景中。假设一个矩形的长为 3x + 2，宽为 x + 1。它的周长是 2(长 + 宽) = 2[(3x + 2) + (x + 1)] = 2(4x + 3) = 8x + 6。这里，在括号内合并同类项简化了计算。

In geometry and physics problems, expressions often appear in combined form. An area problem may require adding the areas of two rectangles: one with area 2x(3x + 1) and another with area x(4x – 2). After expanding both, we collect like terms to express the total area as a single polynomial 6x² + 2x + 4x² – 2x = 10x². The x terms cancel out completely.

在几何和物理问题中，表达式常以组合形式出现。一个面积问题可能需要将两个矩形的面积相加：一个面积为 2x(3x + 1)，另一个面积为 x(4x – 2)。展开两者后，我们合并同类项，将总面积表示为单一多项式 6x² + 2x + 4x² – 2x = 10x²。这里的 x 项完全抵消了。

10. Quick Self-Check | 快速自测

Test your understanding with these short problems. Simplify each expression.

通过以下简短的练习测试你的理解。化简下列各式。

1. 5a + 2a	2. 7x – 3x + 4	3. 2m² + 5m + m²
4. 4(2x + 1) – 3(x + 2)	5. -2(-3x + 5) + 4x	6. 3xy + 4x – 2xy + 5

Answers: 1. 7a. 2. 4x + 4. 3. 3m² + 5m. 4. 8x + 4 – 3x – 6 = 5x – 2. 5. 6x – 10 + 4x = 10x – 10. 6. xy + 4x + 5.

答案：1. 7a。2. 4x + 4。3. 3m² + 5m。4. 8x + 4 – 3x – 6 = 5x – 2。5. 6x – 10 + 4x = 10x – 10。6. xy + 4x + 5。

In question 6, note that 3xy and -2xy are like terms because they share both the variable x and the variable y, each to the first power. However, xy and x are not like terms, so they remain separate in the final answer.

在第 6 题中，注意 3xy 和 -2xy 是同类项，因为它们同时包含变量 x 和变量 y，且都是一次幂。然而，xy 和 x 不是同类项，因此它们在最终答案中保持分离。

Collecting like terms is a fundamental skill that will appear in every algebra exam paper. Master it now, and later topics from expanding brackets to solving simultaneous equations will feel significantly easier.

合并同类项是一项基础技能，出现在每一份代数考试试卷中。现在掌握它，后续从展开括号到求解联立方程等主题都会感觉轻松许多。

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Collecting Like Terms | 合并同类项