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Collecting Like Terms | 合并同类项

2026-08-21 16:17 · 本文部分内容由AI辅助创作

📚 Collecting Like Terms | 合并同类项

In IGCSE algebra, collecting like terms is one of the first and most important simplification skills. It means adding or subtracting terms that have exactly the same variable part, so that an expression becomes shorter and easier to use.

在IGCSE代数中，合并同类项是最先接触也最重要的化简技能之一。它是指将变量部分完全相同的项相加或相减，使表达式更简短、更易于使用。

1. What Are 'Like Terms'? | 什么是“同类项”

In algebra, a term is a single number, a letter, or a product of numbers and letters. Examples of terms include 7, x, 3x, −2y and 5ab.

在代数中，项是单个数字、单个字母，或者数字与字母的乘积。项的例子有 7、x、3x、−2y 和 5ab。

Like terms are terms that have exactly the same variable part. The coefficient, the number in front, can be different. For example, 4x and −2x are like terms, but 4x and 4y are not like terms.

同类项是指变量部分完全相同的项。系数（前面的数字）可以不同。例如，4x 和 −2x 是同类项，但 4x 和 4y 不是同类项。

Like terms / 同类项Not like terms / 不是同类项
4x and −2x4x and 4x²
7y² and 3y²7y and 7xy
5 and −85x and 5

2. The Golden Rule: Same Variable, Same Power | 黄金法则：同变量、同指数

The variable part includes both the letters and their powers. For two terms to be like terms, both must match exactly. The coefficients can be any numbers, including fractions and decimals.

变量部分包括字母及其指数。两个项要成为同类项，这两部分必须完全一致。系数可以是任何数，包括分数和小数。

The golden rule is: same variable, same power. If either the letter or the exponent differs, the terms cannot be combined directly.

黄金法则是：变量相同，指数相同。如果字母或指数有任何不同，就不能直接合并。

× x. Therefore, 2x + 3x² cannot be simplified any further unless you know the value of x.

例如，x 和 x² 看起来相似，但它们表示不同的量：x 是一个 x，而 x² 是 x × x。因此，2x + 3x² 不能再进一步合并，除非你知道 x 的值。

3. How to Collect Like Terms | 如何合并同类项

To collect like terms, follow three steps: first, identify the terms that have matching variable parts; second, group them mentally or by rewriting the expression; third, add or subtract the coefficients while keeping the variable part unchanged.

合并同类项分为三步：首先，找出变量部分相同的项；其次，在脑海中进行分组或把表达式重写；最后，将系数相加或相减，变量部分保持不变。

3x + 2y + 5x − y = (3 + 5)x + (2 − 1)y = 8x + y

This works because the x terms are 3x and 5x, and the y terms are 2y and −y. The coefficients are combined, not the variables.

这样做是因为 x 项是 3x 和 5x，y 项是 2y 和 −y。合并的是系数，而不是变量。

4. Collecting Terms with Powers | 合并含有幂的项

When an expression contains powers, keep each power group separate. Terms like x², x and constants must be collected independently.

当表达式中含有幂时，要保持各组幂分开。x²、x 和常数项必须分别独立合并。

2x² + 3x + 4x² − 5x + 2 = (2+4)x² + (3−5)x + 2 = 6x² − 2x + 2

Notice that the final expression still has three terms because there are three different types of term: quadratic, linear and constant.

注意，最终表达式仍然有三项，因为存在三种不同类型的项：二次项、一次项和常数项。

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5. Working with Several Variables | 处理多个变量

With several variables, treat each variable combination as its own type. For example, ab and a²b are not like terms, because the power of a is different.

对于多个变量，要将每个变量组合视为独立的类型。例如，ab 和 a²b 不是同类项，因为 a 的指数不同。

However, xy and yx are usually treated as like terms because multiplication is commutative. In IGCSE answers, it is normally best to write variables in alphabetical order.

不过，xy 和 yx 通常被视为同类项，因为乘法满足交换律。在 IGCSE 答案中，通常最好按字母顺序书写变量。

4ab + 2a − ab + 3a = (4−1)ab + (2+3)a = 3ab + 5a

6. Dealing with Negative Coefficients | 处理负系数

Negative coefficients often cause errors. Use the sign directly in front of each term. For example, in 5x − 8x, the coefficient of the second term is −8, so the result is −3x.

负系数经常导致错误。要使用每一项前面的符号。例如，在 5x − 8x 中，第二项的系数是 −8，所以结果是 −3x。

5x − 8x = −3x
−2y − 4y = −6y
3x − 7x + 2x = −2x

Imagine the sign as part of the coefficient. This helps when you group and add positive and negative numbers.

把符号看作系数的一部分。这样在分组并相加正负数时会更有帮助。

7. Simplifying Expressions with Brackets First | 先展开括号再合并

If brackets are present, expand them before collecting like terms. Use the distributive law: multiply the term outside the bracket by every term inside.

如果表达式中含有括号，要先展开括号再合并同类项。使用分配律：将括号外的项与括号内的每一项相乘。

2(x + 3) + 3x = 2x + 6 + 3x = 5x + 6

3(2x − 1) − 2(x + 4) = 6x − 3 − 2x − 8 = 4x − 11

Be especially careful with the minus sign in front of a bracket: it changes every sign inside the bracket.

要特别注意括号前的负号：它会改变括号内每一项的符号。

8. Common Mistakes to Avoid | 常见错误与避免

Some common mistakes to avoid:

以下是一些需要避免的常见错误：

9. Exam-Style Practice Questions | 考试风格练习题

Try these IGCSE-style questions. Simplify each expression as much as possible.

尝试以下 IGCSE 风格练习题，尽可能化简每个表达式。