📚 Simplifying Algebraic Expressions (Cambridge KS3 p113) | 化简代数表达式（剑桥KS3数学第113页）
Welcome to this revision guide based on page 113 of the Cambridge KS3 Mathematics textbook. Here we explore the fundamentals of algebraic expressions and learn how to simplify them by collecting like terms, applying the distributive law, using the index laws and substituting values. Mastering these skills is essential for success in algebra at KS3, IGCSE and beyond.
欢迎来到这篇基于剑桥 KS3 数学教材第 113 页的复习指南。我们将系统探索代数表达式的基础知识，学习如何通过合并同类项、运用分配律、使用指数法则以及代入数值来化简表达式。掌握这些技能，是你在 KS3、IGCSE 乃至更高级别代数学习中取得好成绩的关键。
1. What is an Algebraic Expression? | 什么是代数表达式？
example, 3x + 2, 5a − 7b and x² + 2x − 4 are all algebraic expressions. Unlike an equation, an expression does not contain an equals sign. Think of it as a phrase in mathematics that describes a quantity.
代数表达式是由数字、变量（代表未知值的字母）和运算符号（+、−、×、÷）组合而成的。例如，3x + 2、5a − 7b 和 x² + 2x − 4 都是代数表达式。与方程不同，表达式不包含等号。可以把它看作数学中描述某个数量的短语。
3x + 2y − 5
The key vocabulary matters here. A variable is a letter such as x, y or a. A coefficient is the number multiplying a variable. A constant is a number on its own. To simplify an expression means to write it in the shortest, tidiest equivalent form, without changing its value for any value of the letters.
这里有三个关键词必须分清。变量是像 x、y、a 这样的字母；系数是乘在变量前面的数字；常数是不含字母的单独数字。所谓化简表达式，就是把它写成更简短、更整洁的等价形式，同时保证对字母取任何值时表达式的值都不改变。
2. Terms, Coefficients and Constants | 项、系数和常数
Every algebraic expression is made up of terms. A term can be a single number, a single variable, or a number multiplied by one or more variables. For instance, in 4x² + 3x − 7, the terms are 4x², 3x and −7. The sign in front of a term belongs to that term, so we read the third term as negative seven, not as seven.
每个代数表达式都是由项组成的。一个项可以是单独的数字、单独的变量，或者是数字与一个或多个变量的乘积。例如在 4x² + 3x − 7 中，三个项分别是 4x²、3x 和 −7。请注意：项前面的符号属于这个项本身，所以第三项读作"负七"，而不是"七"。
The number multiplying the variable part is the coefficient: 4 is the coefficient of x² and 3 is the coefficient of x. A term with no variable at all is called a constant term (here −7). If a term is written as just x, the coefficient is understood to be 1; if it is written as −x, the coefficient is −1.
乘在变量部分前面的数字就是系数：4 是 x² 的系数，3 是 x 的系数。完全没有字母的项称为常数项（此处是 −7）。如果项写成单独的 x，系数默认为 1；如果写成 −x，系数就是 −1。这一点在合并同类项时非常容易出错，必须牢记。
| Term | 项 | Coefficient | 系数 | Variable part | 变量部分 |
| 4x² | 4 | x² |
| 3x | 3 | x |
| −7 | −7 | none (constant) | 无（常数） |
3. Like Terms and Unlike Terms | 同类项与异类项
Like terms are terms that have exactly the same variable part, including the same power. For example, 2x and 5x are like terms because both contain x to the power of 1. Similarly, 3y² and −y² are like terms, and 7ab and −4ab are like terms. Unlike terms have different variable parts: 2x and 3x² are not like terms, and 4a and 5b are also not like terms.
同类项是指变量部分完全相同的项，包括指数也必须相同。例如 2x 和 5x 是同类项，因为二者都含有 x 的一次方；同样，3y² 和 −y² 是同类项，7ab 和 −4ab 也是同类项。异类项的变量部分不同：2x 与 3x² 不是同类项，4a 与 5b 也不是同类项。
d. So 2x + 5x = 7x, and 3y² − y² = 2y². Never join unlike terms into one term: 2x + 3x² cannot be written as 5x³ or 5x², it simply stays as 2x + 3x².
只有同类项才能通过加法或减法合并。规则很简单：把系数相加减，变量部分保持不变。所以 2x + 5x = 7x，3y² − y² = 2y²。千万不要把异类项硬拼成一项：2x + 3x² 既不能写成 5x³，也不能写成 5x²，它只能保持为 2x + 3x² 的形式。
- Like terms: 6a and −a; 4xy and 9xy; 2m² and 15m²
- Unlike terms: 6a and 6a²; 4xy and 4x; 2m² and 2m³
- 同类项：6a 与 −a；4xy 与 9xy；2m² 与 15m²
- 异类项：6a 与 6a²；4xy 与 4x；2m² 与 2m³
4. Simplifying by Collecting Like Terms | 通过合并同类项化简
Collecting like terms is the most common way to simplify an expression. Work through the expression term by term, group the like terms together, then combine each group. It often helps to underline or highlight each family of terms in a different colour so that nothing is missed, especially when negative signs are involved.
合并同类项是最常用的化简方法。逐个处理表达式中的每一项，把同类项归到一组，然后分别合并每一组。做题时可以用不同颜色的笔把每一类项圈出来，这样不容易漏项，尤其是在出现负号的时候。
4x + 3y + 2x − y = 6x + 2y
idied any further because 6x and 2y are unlike terms. If no like terms remain, the expression is already in its simplest form.
在上面这个例子中，4x + 2x = 6x，3y − y = 2y。注意单独的 y 其实就是 1y，所以 3y − 1y = 2y。答案不能再继续合并，因为 6x 与 2y 是异类项。如果没有可以继续合并的同类项了，那么这个表达式就已经是最简形式。
- 5a + 7a = 12a
- 9b − 4b = 5b
- 6x² + 2x² − x² = 7x²
- 8p + 3 − 2p + 5 = 6p + 8
- 5a + 7a = 12a
- 9b − 4b = 5b
- 6x² + 2x² − x² = 7x²
- 8p + 3 − 2p + 5 = 6p + 8（字母项与常数项分别合并）
5. The Distributive Law and Brackets | 分配律与括号
n we can remove brackets, a process usually called expanding. It also works when the term outside is negative or is itself a variable.
分配律指的是：括号外面的数要乘以括号内的每一项。用符号表示就是 a(b + c) = ab + ac。正是这条法则让我们能够去掉括号，这个过程通常叫做"展开"。当括号外的项是负数、或者本身含有字母时，分配律同样适用。
3(x + 4) = 3x + 12
lecting like terms, for example 5(2a + 3) − 4(a − 1). Expand first, then collect: 10a + 15 − 4a + 4 = 6a + 19. The minus sign in front of the second bracket must multiply both terms inside it.
当括号外的乘数是负数时，括号内每一项的符号都要改变：−2(3x − 5) = −6x + 10。考试中非常常见的题型是把展开与合并同类项结合起来，例如 5(2a + 3) − 4(a − 1)。先展开再合并：10a + 15 − 4a + 4 = 6a + 19。特别注意第二个括号前面的减号必须乘到括号内的两项上，容易漏掉 −4 × (−1) = +4。
- 4(x + 3) = 4x + 12
- 6(2y − 5) = 12y − 30
- −3(x + 2) = −3x − 6
- −2(3x − 5) = −6x + 10
- x(x + 7) = x² + 7x
- 4(x + 3) = 4x + 12
- 6(2y − 5) = 12y − 30
- −3(x + 2) = −3x − 6
- −2(3x − 5) = −6x + 10
- x(x + 7) = x² + 7x（字母乘字母时指数相加）
6. Expanding Double Brackets | 展开双括号
When two brackets are multiplied together, every term in the first bracket must multiply every term in the second. For (x + 3)(x + 5) we get x² + 5x + 3x + 15, which simplifies to x² + 8x + 15. Many students remember this as FOIL: First, Outer, Inner, Last.
当两个括号相乘时，第一个括号中的每一项都要乘以第二个括号中的每一项。对于 (x + 3)(x + 5)，我们得到 x² + 5x + 3x + 15，合并后为 x² + 8x + 15。很多同学用 FOIL 口诀来记忆这个顺序：首项、外项、内项、末项。
(x + 3)(x + 5) = x² + 8x + 15
ish by collecting like terms, because x², x and the constant cannot be merged any further.
遇到负号要格外小心：(x − 4)(x + 2) = x² + 2x − 4x − 8 = x² − 2x − 8。当两个括号完全相同时，(x + 5)² = (x + 5)(x + 5) = x² + 10x + 25。展开完成后一定要合并同类项，因为 x²、x 和常数这三类项之间不能再合并了。
7. The Index Laws for Multiplying and Dividing | 乘除中的指数法则
t them: xᵃ ÷ xᵇ = xᵃ⁻ᵇ. When a power is raised to another power, multiply the indices: (xᵃ)ᵇ = xᵃᵇ.
含有乘方的项同样可以化简。同底数幂相乘，指数相加：xᵃ × xᵇ = xᵃ⁺ᵇ；同底数幂相除，指数相减：xᵃ ÷ xᵇ = xᵃ⁻ᵇ；幂的乘方，指数相乘：(xᵃ)ᵇ = xᵃᵇ。另外，任何非零字母的零次幂都等于 1，即 x⁰ = 1。
ing a whole expression by a single term, divide each term in turn: (8x⁴ + 6x²) ÷ 2x² = 4x² + 3.
记住系数和字母要分开处理：6x × 3x = 18x²，因为 6 × 3 = 18，而 x × x = x²。同理，12x⁵ ÷ 4x² = 3x³。当整个表达式除以单项式时，要逐项相除：(8x⁴ + 6x²) ÷ 2x² = 4x² + 3。
8. Substituting Values into Expressions | 把数值代入表达式
Substitution means replacing each letter with a given number and calculating the result. The most important rule is to follow the order of operations, often remembered as BIDMAS: brackets, indices, division and multiplication, then addition and subtraction.
代入就是把表达式中的每个字母换成给定的数值，然后计算结果。最重要的原则是遵循运算顺序，通常记作 BIDMAS：先算括号，再算乘方，然后算乘除，最后算加减。顺序一旦弄错，答案就会完全不同。
3x² − 2y = 3 × 4² − 2 × (−1) = 48 + 2 = 50 (x = 4, y = −1)
3a = 3 × (−2) = −6. Writing the bracket is the single best habit for avoiding sign errors in substitution questions.
注意 3x² 表示 3 × x²，而不是 (3x)²。当代入的数值为负数时，一定要把它放进括号里：如果 a = −2，那么 a² = (−2)² = 4，而 3a = 3 × (−2) = −6。养成加括号的习惯，是避免代入题符号错误的最好方法。
- If a = 3, then 4 + 2a = 4 + 6 = 10
- If x = 5, then 3x − 7 = 15 − 7 = 8
- If m = 4, then 3m² = 3 × 16 = 48
- If b = −2, then b² + 5b = 4 − 10 = −6
- 若 a = 3，则 4 + 2a = 4 + 6 = 10
- 若 x = 5，则 3x − 7 = 15 − 7 = 8
- 若 m = 4，则 3m² = 3 × 16 = 48
- 若 b = −2，则 b² + 5b = 4 − 10 = −6
9. Forming Expressions from Words and Shapes | 由文字与图形列出表达式
Expressions are often hidden inside word problems and geometry questions. A useful technique is to let a letter stand for the unknown quantity, then translate each phrase into algebra step by step. Words such as 'sum', 'product', 'more than' and 'less than' all have direct algebraic meanings.
表达式常常隐藏在应用题和几何题中。一个实用的技巧是：先用一个字母表示未知量，然后一步步把每句文字翻译成代数式。"和"、"积"、"比……多"、"比……少"这些词都有对应的代数表达方式。
ld in seven years, and 3x represents three times her present age. The perimeter of a triangle with sides 3x, 2x + 1 and 5x − 4 is 3x + (2x + 1) + (5x − 4) = 10x − 3.
例如，长为 l、宽为 w 的矩形，周长为 2(l + w)，展开后是 2l + 2w。如果某个女孩现在 x 岁，那么七年后她是 x + 7 岁，而 3x 表示她现年龄的三倍。三边长分别为 3x、2x + 1 和 5x − 4 的三角形，周长是 3x + (2x + 1) + (5x − 4) = 10x − 3。
10. Common Mistakes to Avoid | 常见错误与避坑指南
Most lost marks in this topic come from a handful of repeated errors. Combining unlike terms is the classic one: 3x + 4y is already simplified and cannot become 7xy. Another is dropping a negative sign when expanding a bracket, or forgetting to multiply the second term inside a bracket.
这一部分的失分大多来自几个反复出现的错误。最典型的是把异类项合并：3x + 4y 已经是最简形式，不能写成 7xy。其次是在展开括号时丢掉负号，或者只乘了括号里的第一项而忘了第二项。
- Wrong: 2x + 3x² = 5x² | Correct: 2x + 3x² stays as it is
- Wrong: −3(x − 4) = −3x − 12 | Correct: −3x + 12
- Wrong: 3x² means (3x)² | Correct: 3x² means 3 × x²
- Wrong: x + x = x² | Correct: x + x = 2x
- Wrong: x × x = 2x | Correct: x × x = x²
- 错误：2x + 3x² = 5x² | 正确：2x + 3x² 保持不变
- 错误：−3(x − 4) = −3x − 12 | 正确：−3x + 12
- 错误：3x² 就是 (3x)² | 正确：3x² 表示 3 × x²
- 错误：x + x = x² | 正确：x + x = 2x
- 错误：x × x = 2x | 正确：x × x = x²
Always check whether your answer can be simplified further, and always show your working line by line. Examiners award method marks even when an arithmetic slip happens at the end.
做完之后一定要检查答案是否还能继续化简，并且养成一行一行写过程的习惯。即使最后一步出现计算失误，阅卷老师仍会根据过程给出方法分。
11. Worked Examples | 例题精讲
+ 10y. Example 2: Expand and simplify 4(2x − 3) + 5(x + 2) = 8x − 12 + 5x + 10 = 13x − 2.
例 1：化简 8x + 3y − 2x + 7y。把含 x 的项和含 y 的项分别归组：8x − 2x = 6x，3y + 7y = 10y，答案是 6x + 10y。例 2：展开并化简 4(2x − 3) + 5(x + 2) = 8x − 12 + 5x + 10 = 13x − 2。
× (−2) = 50 − 6 = 44. Example 4: Expand (x + 2)(x + 6) = x² + 6x + 2x + 12 = x² + 8x + 12.
例 3：当 a = 5、b = −2 时，求 2a² + 3b 的值。把数值加括号代入：2 × 25 + 3 × (−2) = 50 − 6 = 44。例 4：展开 (x + 2)(x + 6) = x² + 6x + 2x + 12 = x² + 8x + 12。
12. Practice Questions and Answers | 练习与答案
Try these questions without looking at the answers first. They cover every skill in this guide: collecting like terms, expanding brackets, index laws and substitution. Aim to finish them in about fifteen minutes, then check your working carefully.
请先不要看答案，独立完成下面这些题目。它们覆盖了本篇指南的全部技能：合并同类项、展开括号、指数法则和代入求值。建议在十五分钟内完成，然后仔细核对每一步过程。
- Q1. Simplify 7x + 4x − 3x
- Q2. Simplify 9a + 6b − 4a − 2b
- Q3. Expand 5(3y − 2)
- Q4. Expand and simplify 6(x + 1) − 2(x − 3)
- Q5. Evaluate 4x − 5 when x = 3
- Q6. Evaluate 3m² when m = 4
- Q7. Expand (x + 2)(x + 6)
- Q8. Simplify 10x⁶ ÷ 5x²
- 第 1 题：化简 7x + 4x − 3x
- 第 2 题：化简 9a + 6b − 4a − 2b
- 第 3 题：展开 5(3y − 2)
- 第 4 题：展开并化简 6(x + 1) − 2(x − 3)
- 第 5 题：当 x = 3 时，求 4x − 5 的值
- 第 6 题：当 m = 4 时，求 3m² 的值
- 第 7 题：展开 (x + 2)(x + 6)
- 第 8 题：化简 10x⁶ ÷ 5x²
f you got all eight correct, you have secure control of the material on page 113 and you are ready to move on to solving equations.
答案：第 1 题 8x。第 2 题 5a + 4b。第 3 题 15y − 10。第 4 题 6x + 6 − 2x + 6 = 4x + 12。第 5 题 12 − 5 = 7。
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