📚 Simplifying Algebraic Expressions: A Complete KS3 and IGCSE Guide | 代数表达式化简：KS3 与 IGCSE 完整指南
In KS3 Cambridge Mathematics and IGCSE algebra, simplifying algebraic expressions is one of the most important skills you will ever learn. Almost every later topic depends on it: solving equations, rearranging formulae, working with sequences, and tackling algebraic fractions all begin with the same basic moves. You need to recognise like terms, collect them correctly, expand brackets, factorise simple expressions, and substitute values accurately. This article explains each skill step by step with clear worked examples, a vocabulary table, a list of common mistakes, and exam-style practice.
在 KS3 剑桥数学与 IGCSE 代数中，化简代数表达式是最重要的技能之一。后续几乎所有主题都依赖它：解方程、变形公式、处理数列以及代数分式，都从同样的基本操作开始。你需要识别同类项、正确合并同类项、展开括号、因式分解简单表达式，并准确代入数值。本文将逐步讲解每项技能，配以清晰的例题、词汇表、常见错误清单以及考试风格的练习。
1. Terms, Coefficients and Constants | 项、系数与常数
An algebraic expression is a combination of numbers, variables and operations. For example, 5x + 3y − 7 is an expression. Each part separated by a '+' or a '−' sign is called a term. In 5x + 3y − 7, the terms are 5x, 3y and −7, so there are three terms in total.
代数表达式是数字、变量和运算的组合。例如，5x + 3y − 7 是一个表达式。由 '+' 或 '−' 号分隔的每一部分称为一项。在 5x + 3y − 7 中，项为 5x、3y 和 −7，因此共有三项。
The number multiplying a variable is called the coefficient. In the term 5x, the coefficient is 5. If a term is written simply as x, the coefficient is 1, because x means 1x. If a term is written as −x, the coefficient is −1. A term without any variable, such as −7, is called a constant term, because its value never changes.
乘以变量的数称为系数。在项 5x 中，系数为 5。如果一项只写作 x，其系数为 1，因为 x 就是 1x。如果一项写作 −x，其系数为 −1。不含任何变量的项（如 −7）称为常数项，因为它的值永远不变。
Here are the main vocabulary words you need to know:
以下是你需要掌握的主要词汇：
- Variable: a symbol, such as x or y, that represents an unknown number.
- 变量：一个符号，如 x 或 y，代表未知数。
- Coefficient: the number multiplied by the variable.
- 系数：乘以变量的数。
- Constant: a fixed number, such as 5 or −3.
- 常数：一个固定的数，如 5 或 −3。
- Term: a single number, a variable, or a product of numbers and variables.
- 项：单个数字、变量，或数字与变量的乘积。
- Expression: a collection of terms joined by '+' or '−' signs, with no equals sign.
- 表达式：由 '+' 或 '−' 号连接的若干项，不含等号。
The table below breaks one expression into its parts so you can see the structure clearly.
下表把一个表达式拆解成各个部分，让你清楚看到结构。
| Term 项 | Coefficient 系数 | Variable part 变量部分 |
| 5x | 5 | x |
| 3y | 3 | y |
| −7 | −7 | none (constant 常数) |
2. Collecting Like Terms | 合并同类项
Like terms have exactly the same variable part, including the same power. For instance, 4x and 9x are like terms, but 4x and 4x² are not like terms, because the powers of x are different. You can add or subtract like terms by adding or subtracting their coefficients, while the variable part stays unchanged.
同类项具有完全相同的变量部分，包括相同的幂。例如，4x 和 9x 是同类项，但 4x 和 4x² 不是同类项，因为 x 的幂不同。你可以通过加减系数来合并同类项，而变量部分保持不变。
Example: Simplify 7a + 3b − 2a + 5b.
示例：化简 7a + 3b − 2a + 5b。
7a + 3b − 2a + 5b = (7a − 2a) + (3b + 5b) = 5a + 8b
Notice that the a-terms and the b-terms are collected separately. The sign in front of a term always travels with that term, so −2a is collected as negative 2a.
注意 a 项与 b 项要分开合并。项前面的符号始终跟随该项，所以 −2a 作为负的 2a 来合并。
Another example with a constant: 4x + 9 − 2x + 3.
再看一个含常数项的例子：4x + 9 − 2x + 3。
4x + 9 − 2x + 3 = (4x − 2x) + (9 + 3) = 2x + 12
A useful routine is to underline each family of like terms in a different way, or to rewrite the expression with the like terms grouped together, before you do any arithmetic.
一个实用做法是：用不同方式标注每一组同类项，或者先把同类项分组重写，然后再做运算。
3. Working with Negative Coefficients | 处理负系数
Negative signs cause more lost marks than almost anything else in algebra. The key rule is simple: a minus sign belongs to the term that follows it. So in 8x − 5 − 3x, the terms are 8x, −5 and −3x, which gives 5x − 5.
负号在代数中造成的失分几乎比其他任何问题都多。关键规则很简单：减号属于它后面的那一项。因此在 8x − 5 − 3x 中，项是 8x、−5 和 −3x，化简得 5x − 5。
m is subtracted, the result becomes less negative, or positive: 5y − (−3y) = 5y + 3y = 8y.
当两个负项相加时，结果更负：−4y + (−2y) = −6y。当减去一个负项时，结果负得更少，甚至变为正：5y − (−3y) = 5y + 3y = 8y。
Example: Simplify 6 − 2m − 4 + 9m.
示例：化简 6 − 2m − 4 + 9m。
6 − 2m − 4 + 9m = (6 − 4) + (9m − 2m) = 2 + 7m
It is perfectly acceptable to write the answer as 7m + 2. Maths convention puts the variable term first, but both forms are correct and both gain full marks in exams.
把答案写成 7m + 2 完全可以。数学惯例是把含变量的项写在前面，但两种形式都正确，考试中都能得满分。
4. Multiplying and Dividing Terms: the Index Laws | 项的乘除与指数法则
Unlike adding, multiplying terms does not require like terms. You multiply the coefficients and multiply the variables separately. To multiply powers of the same letter, add the indices; to divide, subtract the indices.
与加法不同，乘法不需要同类项。你把系数相乘，把变量单独相乘。同字母的幂相乘时指数相加；相除时指数相减。
xᵃ × xᵇ = xᵃ⁺ᵇ xᵃ ÷ xᵇ = xᵃ⁻ᵇ (xᵃ)ᵇ = xᵃᵇ
Example: Simplify 3x × 4x.
示例：化简 3x × 4x。
3x × 4x = (3 × 4) × (x × x) = 12x²
Example: Simplify 12x⁵ ÷ 4x².
示例：化简 12x⁵ ÷ 4x²。
12x⁵ ÷ 4x² = (12 ÷ 4) × x⁵⁻² = 3x³
going on to claim x² × x³ = x⁶. Always remember: multiplying the letters adds the powers.
常见错误是把指数相乘而不是相加，例如把 x² × x³ 写成 x⁶。请牢记：字母相乘时指数相加。
5. Expanding Single Brackets | 展开单括号
Expanding means removing the brackets by multiplying every term inside the bracket by the term outside. This is the distributive law. After expanding, always collect any like terms that appear.
展开就是通过把括号外的一项与括号内每一项相乘来去掉括号。这就是分配律。展开之后，务必合并出现的同类项。
Example: Expand 3(2x + 5).
示例：展开 3(2x + 5)。
3(2x + 5) = 3 × 2x + 3 × 5 = 6x + 15
Example with a negative multiplier: Expand −2(x − 4).
含负系数的示例：展开 −2(x − 4)。
−2(x − 4) = −2 × x + (−2) × (−4) = −2x + 8
Notice that the second sign flips from minus to plus, because a negative multiplied by a negative gives a positive. Forgetting this change of sign is the single most common expansion mistake.
注意第二个符号由减变加，因为负数乘负数得正数。忘记这个变号是展开中最常见的错误。
Example: Expand and simplify 4(x + 3) + 2(x − 5).
示例：展开并化简 4(x + 3) + 2(x − 5)。
4(x + 3) + 2(x − 5) = 4x + 12 + 2x − 10 = 6x + 2
6. Expanding Double Brackets | 展开双括号
To expand two brackets, multiply every term in the first bracket by every term in the second bracket. With (x + a)(x + b) there are four products in total, so the method is often called FOIL: First, Outer, Inner, Last.
展开两个括号时，把第一个括号中的每一项与第二个括号中的每一项相乘。对于 (x + a)(x + b)，一共有四个乘积，所以这种方法常被称为 FOIL：首项、外项、内项、末项。
Example: Expand (x + 3)(x + 5).
示例：展开 (x + 3)(x + 5)。
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
The two middle terms, 5x and 3x, are like terms, so they combine to give 8x. If you do not collect them, you lose a mark even if your four products are correct.
中间两项 5x 与 3x 是同类项，合并得 8x。如果不合并，即使四个乘积都正确，也会失分。
Example with a negative constant: Expand (x − 4)(x + 7).
含负常数项的示例：展开 (x − 4)(x + 7)。
(x − 4)(x + 7) = x² + 7x − 4x − 28 = x² + 3x − 28
of the two numbers and the coefficient of x equals their sum: −4 × 7 = −28 and −4 + 7 = 3.
快速检验方法：常数项等于两数之积，x 的系数等于两数之和：−4 × 7 = −28，−4 + 7 = 3，与结果一致。
7. Factorising Simple Expressions | 因式分解简单表达式
Factorising is the reverse of expanding. To factorise a simple expression, take out the highest common factor of all the terms and place it outside a bracket. The remaining factors go inside the bracket.
因式分解是展开的逆运算。要分解一个简单表达式，先提取所有项的最高公因式放到括号外，剩下的因式留在括号内。
Example: Factorise 6x + 9.
示例：分解 6x + 9。
6x + 9 = 3(2x + 3)
Example: Factorise x² + 5x.
示例：分解 x² + 5x。
x² + 5x = x(x + 5)
Example with two different letters: Factorise 8ab − 12a.
含两个字母的示例：分解 8ab − 12a。
8ab − 12a = 4a(2b − 3)
Always check your answer by expanding it again. If 4a(2b − 3) expands back to 8ab − 12a, your factorisation is correct. This takes five seconds and catches almost every error.
一定要把答案重新展开来检验。如果 4a(2b − 3) 展开后回到 8ab − 12a，说明分解正确。这一步只需五秒，却能发现几乎所有错误。
8. Substitution | 代入数值
Substitution means replacing the letters in an expression with given numbers and then working out the value. Replace each letter with its value in brackets, then follow the order of operations: indices, then multiplication and division, then addition and subtraction.
代入就是用给定数值替换表达式中的字母，再算出结果。把每个字母用它带括号的数值替换，然后遵守运算顺序：先乘方，再乘除，最后加减。
Example: Find the value of 4x − 3y + 2 when x = 3 and y = −2.
示例：当 x = 3、y = −2 时，求 4x − 3y + 2 的值。
4(3) − 3(−2) + 2 = 12 + 6 + 2 = 20
Example with a power: Find the value of 2x² + x when x = 3.
含乘方的示例：当 x = 3 时，求 2x² + x 的值。
2(3)² + 3 = 2 × 9 + 3 = 18 + 3 = 21
in brackets, so −3(−2) is clearly +6, while −3 × −2 written loosely often loses its sign.
要避免两个陷阱：第一，先对数字平方再乘系数，所以 x = 3 时 2x² 等于 18，不是 36。第二，负值一定要加括号，这样 −3(−2) 清楚地等于 +6，而不加括号时符号很容易丢失。
9. Simplifying Algebraic Fractions | 化简代数分式
An algebraic fraction is simplified by dividing the numerator and denominator by their common factors. If every term on top shares a factor with the bottom, you can cancel it. Division of single terms uses the index law you met earlier.
化简代数分式的方法是：用分子与分母的公因式相除。如果分子每一项都与分母有公因式，就可以约去。单项式的除法使用前面学过的指数法则。
Example: Simplify (6x + 9) ÷ 3.
示例：化简 (6x + 9) ÷ 3。
(6x + 9) ÷ 3 = 6x ÷ 3 + 9 ÷ 3 = 2x + 3
Example: Simplify (x² + 5x) ÷ x for x ≠ 0.
示例：当 x ≠ 0 时，化简 (x² + 5x) ÷ x。
(x² + 5x) ÷ x = x² ÷ x + 5x ÷ x = x + 5
Example: Simplify 12x⁵ ÷ 4x².
示例：化简 12x⁵ ÷ 4x²。
12x⁵ ÷ 4x² = 3x³
One warning: you can only cancel a factor, never a term. In (x + 5) ÷ x you may not cancel the x in the numerator with the x in the denominator, because the x on top is part of a sum, not a factor of the whole numerator.
一个警告：只能约去因式，绝不能约去项。在 (x + 5) ÷ x 中，不能把分子里的 x 与分母的 x 约去，因为分子上的 x 只是一个加数，并不是整个分子的因式。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Most lost marks in simplification questions come from a small number of repeated errors. Recognising them in advance is the fastest way to improve your accuracy.
化简题中的失分大多来自少数重复性错误。提前识别这些错误，是提高准确率最快的办法。
- Adding unlike terms: 4x + 4x² is not 8x³ and not 8x²; the expression is already as simple as possible.
- 错误地合并不同类项：4x + 4x² 不等于 8x³，也不等于 8x²；该表达式已经是最简形式。
- Forgetting the sign: −3(x − 2) = −3x + 6, not −3x − 6.
- 忘记变号：−3(x − 2) = −3x + 6，而不是 −3x − 6。
- Multiplying indices when you should add: x² × x³ = x⁵, not x⁶.
- 该加指数时却相乘：x² × x³ = x⁵，不是 x⁶。
- Only multiplying the first term in a bracket: 5(x + 2) is 5x + 10, not 5x + 2.
- 只乘括号中的第一项：5(x + 2) 等于 5x + 10，不是 5x + 2。
- Cancelling terms instead of factors in algebraic fractions.
- 在代数分式中约去项而不是因式。
- Ignoring order of operations during substitution.
- 代入时忽略运算顺序。
to the original expression and into your simplified answer. If the two values match, your simplification is almost certainly correct.
一个可靠的检查习惯是：把一个简单的数（如 x = 2）分别代入原表达式和化简后的答案。若两个值相等，你的化简几乎可以肯定是对的。
11. Worked Exam-Style Questions | 考试风格例题精讲
Question 1: Simplify 5x + 7y − 2x + 3y − 4.
第 1 题：化简 5x + 7y − 2x + 3y − 4。
= (5x − 2x) + (7y + 3y) − 4 = 3x + 10y − 4
Question 2: Expand and simplify 3(2x − 1) − 2(x − 4).
第 2 题：展开并化简 3(2x − 1) − 2(x − 4)。
= 6x − 3 − 2x + 8 = 4x + 5
Question 3: Factorise fully 10x² + 15x.
第 3 题：完整分解 10x² + 15x。
= 5x(2x + 3)
Question 4: Expand (x + 6)(x − 2) and state the coefficient of x.
第 4 题：展开 (x + 6)(x − 2)，并写出 x 的系数。
= x² − 2x + 6x − 12 = x² + 4x − 12, so the coefficient of x is 4
Question 5: Given that x = 4 and y = −1, work out the value of 3x² − 2xy.
第 5 题：已知 x = 4、y = −1，求 3x² − 2xy 的值。
= 3(4)² − 2(4)(−1) = 48 + 8 = 56
In each case, write one clear line of working for every step. Examiners award method marks even when the final answer is wrong, but only if your reasoning is visible.
每一题都要为每个步骤写一行清晰的运算过程。即使最终答案错误，只要推理可见，考官仍会给出方法分。
12. Summary and Revision Checklist | 总结与复习清单
Simplifying algebraic expressions is built from a small set of rules that you can apply in any order. Identify the terms, group the like terms, deal with the brackets, then check your answer by expanding or substituting.
化简代数表达式建立在一小组规则之上，你可以按任意顺序运用它们：识别各项、分组同类项、处理括号，最后通过展开或代入来检验答案。
- Know the difference between a term, a coefficient and a constant.
- 分清项、系数与常数的区别。
- Only collect terms with identical variable parts and identical powers.
- 只合并变量部分与幂完全相同的项。
- Add indices when multiplying powers, subtract when dividing.
- 幂相乘时指数相加，相除时指数相减。
- Multiply every term inside a bracket by the term outside.
- 括号外的项要乘括号内的每一项。
- Combine the two middle terms after expanding double brackets.
- 展开双括号后要合并中间两项。
- Factorising is expanding in reverse; always check by re-expanding.
- 因式分解是展开的逆过程；务必重新展开检验。
- Put negative values in brackets during substitution.
- 代入时负值要加括号。
- Cancel factors in fractions, never terms.
- 分式中约去因式，绝不约去项。
If you can do all eight of these reliably, you are ready for the simplification questions in KS3 Cambridge Mathematics, IGCSE algebra papers and beyond. Practise five mixed questions a day for two weeks and your accuracy will improve noticeably.
如果你能稳定完成以上八点，就已为 KS3 剑桥数学、IGCSE 代数试卷以及更高阶段的化简题做好准备。每天练习五道混合题，坚持两周，你的准确率会明显提升
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