Collecting Like Terms | 合并同类项

📚 Collecting Like Terms | 合并同类项

In algebra, expressions often contain several terms that can be simplified. Collecting like terms is the process of grouping together terms that have the same variable part, so that the expression becomes shorter and easier to work with.

在代数中，表达式通常包含多个可以化简的项。合并同类项是将具有相同变量部分的项分组在一起的过程，从而使表达式更简短、更易于处理。

1. What Are Terms? | 什么是项

A term is a single number, a variable, or a product of numbers and variables. For example, in the expression \(3x + 5y – 2\), the terms are \(3x\), \(5y\), and \(-2\).

项是单个数字、单个变量或数字与变量的乘积。例如，在表达式 \(3x + 5y – 2\) 中，各项分别为 \(3x\)、\(5y\) 和 \(-2\)。

Each term has a sign: the sign before it belongs to that term.

每一项都有自己的符号：该项前面的符号属于该项。
A term with no variable is called a constant term.

不含变量的项称为常数项。
The number part of a term is called the coefficient. In \(\,4x\), the coefficient is 4.

项中的数字部分称为系数。在 \(\,4x\) 中，系数是 4。

2. Like Terms vs Unlike Terms | 同类项与非同类项

Like terms have exactly the same variable part: the same variables raised to the same powers. The coefficients can be different.

同类项具有完全相同的变量部分：相同的变量、相同的指数。系数可以不同。

Like terms: \(3x\) and \(-5x\); \(2y^2\) and \(7y^2\); \(4ab\) and \(-ab\).

同类项：\(3x\) 与 \(-5x\)；\(2y^2\) 与 \(7y^2\)；\(4ab\) 与 \(-ab\)。
Unlike terms: \(3x\) and \(3y\) (different variables); \(2a^2\) and \(2a\) (different powers).

非同类项：\(3x\) 与 \(3y\)（变量不同）；\(2a^2\) 与 \(2a\)（指数不同）。

\(3x\) and \(-2x\) are like terms; \(3x\) and \(-2x^2\) are not.

\(3x\) 与 \(-2x\) 是同类项；\(3x\) 与 \(-2x^2\) 不是。

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

Simplifying expressions makes them easier to evaluate, compare, and manipulate in equations. It is similar to combining identical objects: 3 apples plus 2 apples give 5 apples, but 3 apples plus 2 bananas cannot be combined into one simple fruit.

化简表达式使其更容易求值、比较以及在方程中操作。这类似于合并相同的物体：3 个苹果加 2 个苹果得到 5 个苹果，但 3 个苹果加 2 个香蕉不能合并成一种简单的单一水果。

For example:

例如：

\(x + x + x = 3x\)

Here, \(x\) acts like a common object; adding three of them gives three \(x\)’s.

这里，\(x\) 像一个共同的物体；把三个这样的物体相加就得到三个 \(x\)。

4. The Distributive Property | 分配律

Before collecting like terms, you may need to expand brackets using the distributive property:

在合并同类项之前，你可能需要用分配律展开括号：

\(a(b + c) = ab + ac\)

When a negative sign is in front of the bracket, every term inside changes sign.

当括号前面有负号时，括号内每一项的符号都要改变。

Expand: \(2(3x + 4) = 6x + 8\).

展开：\(2(3x + 4) = 6x + 8\)。
Expand: \(- (2x – 5) = -2x + 5\).

展开：\(- (2x – 5) = -2x + 5\)。

5. Collecting Like Terms with Addition and Subtraction | 含加减法的合并同类项

To collect like terms, combine the coefficients of terms that have the same variable part, keeping the variable part unchanged.

要合并同类项，将具有相同变量部分的项的系数相加或相减，变量部分保持不变。

Worked example:

例题：

Simplify \(5x + 3y – 2x + y\).

化简 \(5x + 3y – 2x + y\)。

Group the \(x\)-terms: \(5x – 2x = 3x\).

把含 \(x\) 的项分组：\(5x – 2x = 3x\)。
Group the \(y\)-terms: \(3y + y = 4y\).

把含 \(y\) 的项分组：\(3y + y = 4y\)。
Final answer: \(3x + 4y\).

最终答案：\(3x + 4y\)。

Note that \(3x\) and \(4y\) cannot be combined further because they are unlike terms.

注意 \(3x\) 和 \(4y\) 不能再合并，因为它们是非同类项。

6. Combining Like Terms with Powers | 合并含幂的同类项

Only terms with the same power of the same variable can be combined. For example, \(x^2\) and \(x\) are not like terms.

只有同一个变量的同次幂项才能合并。例如，\(x^2\) 和 \(x\) 不是同类项。

\(3x^2 + 2x – x^2 + 5x = (3 – 1)x^2 + (2 + 5)x = 2x^2 + 7x\)

Notice that \(2x^2\) and \(7x\) are unlike terms, so the expression stops here.

注意 \(2x^2\) 和 \(7x\) 是非同类项，因此表达式在这里停止合并。

7. Simplifying Expressions with Brackets | 含括号表达式的化简

When an expression contains brackets, always expand them first, then collect like terms.

当表达式包含括号时，总是先展开括号，然后再合并同类项。

Worked example:

例题：

Simplify \(3(x + 2) + 2(2x – 1)\).

化简 \(3(x + 2) + 2(2x – 1)\)。

Expand: \(3x + 6 + 4x – 2\).

展开：\(3x + 6 + 4x – 2\)。
Collect \(x\)-terms: \(3x + 4x = 7x\).

合并含 \(x\) 的项：\(3x + 4x = 7x\)。
Collect constants: \(6 – 2 = 4\).

合并常数项：\(6 – 2 = 4\)。
Final answer: \(7x + 4\).

最终答案：\(7x + 4\)。

If there is a minus sign before a bracket, distribute the negative sign to every term inside.

如果括号前有减号，则要将负号分配给括号内的每一项。

8. Multiplying Terms | 项的乘法

When multiplying terms, multiply the coefficients together and multiply the variable parts separately using index laws.

相乘时，先将系数相乘，再分别用指数法则处理变量部分。

\(2x \cdot 3x = (2 \times 3)(x \cdot x) = 6x^2\)

\(2x \cdot 3x = (2 \times 3)(x \cdot x) = 6x^2\)
\(4a^2 \cdot 3a = 12a^3\) (because \(a^2 \cdot a = a^{2+1} = a^3\))

\(4a^2 \cdot 3a = 12a^3\)（因为 \(a^2 \cdot a = a^{2+1} = a^3\)）

After multiplying, check whether any resulting terms are like terms and can be combined.

相乘之后，检查结果中是否有同类项可以合并。

9. Common Mistakes | 常见错误

Students often make simple errors when collecting like terms. Here are the most common ones to avoid.

学生在合并同类项时经常犯一些简单错误。以下是需要避免的常见错误。

Mistake / 错误	Correct / 正确	Explanation / 解释
\(x + x = x^2\)	\(x + x = 2x\)	Adding terms is not the same as multiplying them.
\(3x + 2y = 5xy\)	Cannot be simplified.	Different variables are unlike terms.
\(2x^2 + 3x = 5x^3\)	Cannot be simplified.	Different powers of \(x\) are unlike terms.
\(-(x + 3) = -x + 3\)	\(-(x + 3) = -x – 3\)	The minus sign applies to every term inside.

Always double-check the sign and the power of each term before combining.

在合并之前，请务必反复检查每一项的符号和指数。

10. Practice Questions | 练习题

Try these problems on your own, then check the answers below.

请先尝试独立完成以下题目，再对照下方的答案。

Simplify \(7a + 3b – 2a + b\).

化简 \(7a + 3b – 2a + b\)。
Simplify \(4x^2 – x + 3x^2 + 5x\).

化简 \(4x^2 – x + 3x^2 + 5x\)。
Simplify \(2(3x – 1) – (x + 4)\).

化简 \(2(3x – 1) – (x + 4)\)。
Simplify \(5p + 2q – 3p + q – 4q\).

化简 \(5p + 2q – 3p + q – 4q\)。

Answers:

答案：

1. \(5a + 4b\) 2. \(7x^2 + 4x\) 3. \(5x – 6\) 4. \(2p – q\)

11. Summary | 总结

Collecting like terms is a fundamental skill in algebra. Remember the golden rules: identify the variable and its exponent, group only terms that match completely, and always follow the sign of each term.

合并同类项是代数中的基本技能。记住黄金法则：识别变量及其指数，只合并完全匹配的项，并且始终注意每一项的符号。

Like terms have the same variable(s) raised to the same power(s).

同类项是具有相同变量和相同指数的项。
Combine only the coefficients; the variable part stays unchanged.

只合并系数；变量部分保持不变。
Expand brackets first when they appear in an expression.

当表达式中出现括号时，先展开括号。
Be careful with negative signs and powers.

小心负号和指数。

Once you master these rules, simplifying algebraic expressions becomes quick and reliable.

一旦掌握这些规则，化简代数表达式将变得快速而可靠。

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Collecting Like Terms | 合并同类项