📚 Algebra: Collecting Like Terms | 代数：合并同类项
In the study of algebra, expressions such as 3x + 2x = 5x may seem simple, yet they rest on a foundational skill: collecting like terms. This technique is the first step toward simplifying algebraic expressions, solving equations, and understanding more advanced topics like factorisation and quadratic functions.
在代数的学习中，像 3x + 2x = 5x 这样的表达式看似简单，却依赖一项基础技能：合并同类项。这项技巧是化简代数表达式、解方程以及理解因式分解和二次函数等更高阶课题的第一步。
1. Understanding Algebraic Expressions and Terms | 理解代数表达式与项
An algebraic expression is a combination of numbers, letters, and operation signs, such as 3x + 4 or 2y² − 5y + 1. Each part separated by a plus or minus sign is called a term.
代数表达式由数字、字母和运算符号组合而成，例如 3x + 4 或 2y² − 5y + 1。被加号或减号分隔的每一部分被称为“项”。
- Term (项): 4x, 7, −3ab.
- Coefficient (系数): the number multiplying a variable, e.g. 4 in 4x.
- Constant (常数项): a term without a variable, e.g. 7 in 3x + 7.
In the term −3xy, the coefficient is −3 and the variable part is xy. Notice that the sign of the term is included in the coefficient.
在项 −3xy 中，系数是 −3，变量部分是 xy。注意项的符号包含在系数之中。
| Expression | Terms | Coefficients |
|---|---|---|
| 5x − 3y + 2 | 5x, −3y, +2 | 5, −3, none |
| 4a² − a + 7 | 4a², −a, +7 | 4, −1, none |
2. Defining Like Terms | 同类项的定义
Like terms are terms that have exactly the same variable part, including the same powers. Only like terms can be added or subtracted together.
同类项是指变量部分完全相同的项，包括相同的幂次。只有同类项才能进行加减合并。
For example, 4x and −2x are like terms because both contain x¹. However, 4x² and 4x are not like terms because their powers differ. Similarly, 3xy and 3xy² are not like terms.
例如，4x 与 −2x 是同类项，因为两者都包含 x¹。然而 4x² 与 4x 不是同类项，因为它们的幂次不同。同理，3xy 与 3xy² 也不是同类项。
- 5x and −2x → like terms → 同类项
- 3x² and 7x² → like terms → 同类项
- 4xy and 4x → not like terms → 不是同类项
- 2 and −9 → like terms (constants) → 常数项互为同类项
Constant terms are always like terms with each other, so numbers can always be combined.
常数项之间互为同类项，因此数字之间总是可以合并的。
3. Collecting Like Terms: Addition and Subtraction | 合并同类项：加法与减法
To collect like terms, add or subtract their coefficients while keeping the variable part unchanged. This is the most direct application of what you have learned.
合并同类项时，只需将系数相加或相减，变量部分保持不变。这是上述概念最直接的应用。
4x + 3x = (4 + 3)x = 7x
9y − 4y = (9 − 4)y = 5y
When simplifying an entire expression, group the like terms first, then combine each group. Take 7a + 5b − 2a + 3b: group the a terms and the b terms to get (7a − 2a) + (5b + 3b) = 5a + 8b.
化简整个表达式时，先找出同类项并分组，再逐组合并。例如 7a + 5b − 2a + 3b：将含 a 的项和含 b 的项分组，得到 (7a − 2a) + (5b + 3b) = 5a + 8b。
4. Multiplication Rules: Adding Indices | 乘法法则：指数相加
When multiplying two algebraic terms, multiply the coefficients first, then multiply the variable parts. For the same base variable, add its indices.
两个代数项相乘时，先乘系数，再乘变量部分。若底数变量相同，则将其指数相加。
am × an = am+n
For example, 3x² × 2x³ = 6x⁽²⁺³⁾ = 6x⁵. The coefficient 3 × 2 gives 6, and the indices 2 + 3 give 5.
例如，3x² × 2x³ = 6x⁽²⁺³⁾ = 6x⁵。系数 3 × 2 得 6，指数 2 + 3 得 5。
If the variables are different, keep them separate: 2a × 3b = 6ab. You cannot combine a and b.
如果变量不同，则分开书写：2a × 3b = 6ab。a 和 b 不能合并。
5. Division Rules: Subtracting Indices | 除法法则：指数相减
When dividing two algebraic terms, divide the coefficients and subtract the indices of the same base variable.
两个代数项相除时，系数相除，同底数变量的指数相减。
am ÷ an = am−n
For example, 6x⁵ ÷ 2x² = 3x³, because 6 ÷ 2 = 3 and 5 − 2 = 3. This process is often called “cancelling down”.
例如，6x⁵ ÷ 2x² = 3x³，因为 6 ÷ 2 = 3，且 5 − 2 = 3。这个过程常被称为“约分”。
If the index becomes negative, the result is written as a fraction. For instance, x² ÷ x⁵ = x⁻³ = 1/x³.
如果指数变为负数，结果可写成倒数形式。例如 x² ÷ x⁵ = x⁻³ = 1/x³。
6. The Distributive Law and Expanding Brackets | 分配律与展开括号
Expanding brackets is closely connected to collecting like terms, because once brackets are removed, you often need to combine the resulting like terms.
展开括号与合并同类项密切相关，因为去掉括号后，通常还需要合并产生的同类项。
a(b + c) = ab + ac
For example, 3(2x + 5) = 6x + 15. With a negative sign, every term inside the brackets changes sign: −2(x − 3) = −2x + 6.
例如，3(2x + 5) = 6x + 15。当括号前有负号时，括号内每一项都要变号：−2(x − 3) = −2x + 6。
When brackets are nested, expand from the innermost set outwards, then collect like terms.
遇到多重括号时，从最内层开始展开，然后合并同类项。
7. Combining Fractional Coefficients | 合并分数系数
Fractional coefficients are combined using the same rules, but the fractions must be written over a common denominator first.
分数系数的合并遵循相同规则，但先将分数化为公共分母。
½x + ⅓x = (3/6 + 2/6)x = 5/6x
Similarly, ¾y − ¼y = ½y, because ¾ − ¼ = 2/4 = ½.
同理，¾y − ¼y = ½y，因为 ¾ − ¼ = 2/4 = ½。
When the coefficient is a mixed number, convert it to an improper fraction before combining. For example, 1½x + ½x = 2x.
系数为带分数时，先化为假分数再合并。例如 1½x + ½x = 2x。
8. The Reverse of Collecting: Factorisation | 合并的逆运算：因式分解
Factorisation is the reverse process of collecting like terms. It means writing an expression as a product of a common factor and another expression.
因式
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