Simplifying Algebraic Expressions | 代数式化简

📚 Simplifying Algebraic Expressions | 代数式化简

Algebra is the branch of mathematics where letters and symbols are used to represent numbers in expressions, equations and formulae. In this article, we focus on simplifying algebraic expressions, a core skill in the Cambridge Lower Secondary Mathematics curriculum at KS3.

代数是数学中一门用字母和符号表示数的一支，应用于表达式、方程和公式。本文重点讲解代数式化简，这是剑桥初中数学 KS3 阶段的核心技能。

1. What Is an Algebraic Expression? | 什么是代数式？

An algebraic expression is a combination of numbers, letters, operation signs and sometimes brackets. It does not have an equals sign, so it represents a value that may change depending on the letters.

代数式是由数字、字母、运算符号以及有时出现的括号组合而成的式子。它没有等号，因此代表的值会随着字母的变化而变化。

For example, 3x + 5, 2a − 4b and 7y² + y − 1 are algebraic expressions. Each part of an expression separated by an addition or subtraction sign is called a term.

例如，3x + 5、2a − 4b 和 7y² + y − 1 都是代数式。表达式中被加号或减号分隔的每一部分称为项。

2. Key Terms: Terms, Coefficients and Constants | 项、系数与常数项

In the expression 4x + 3y − 7, there are three terms: 4x, 3y and −7. The number in front of a letter is called the coefficient, so 4 is the coefficient of x and 3 is the coefficient of y.

在表达式 4x + 3y − 7 中，有三项：4x、3y 和 −7。字母前面的数字叫做系数，因此 4 是 x 的系数，3 是 y 的系数。

A term that is just a number, such as −7, is called a constant because its value does not change. The letters x, y, a, b and m are often called variables.

只含数字的项，例如 −7，叫做常数项，因为它的值不改变。字母 x、y、a、b 和 m 通常称为变量。

3. What Are Like Terms? | 什么是同类项？

Like terms are terms that have exactly the same letter part or variable part. The coefficients can be different, but the variables and their powers must match.

同类项是指字母部分完全相同的项。系数可以不同，但变量及其指数必须一致。

5x and −2x are like terms. 是同类项。
3y² and 7y² are like terms. 是同类项。
4ab and −ab are like terms. 是同类项。
3x and 3x² are not like terms because the powers differ. 不是同类项，因为指数不同。
2a and 2b are not like terms because the letters differ. 不是同类项，因为字母不同。

Only like terms can be added or subtracted directly. This idea is the foundation of simplifying algebraic expressions.

只有同类项才能直接相加或相减。这一概念是代数式化简的基础。

4. Combining Like Terms | 合并同类项

To combine like terms, add or subtract their coefficients while keeping the variable part unchanged. This is similar to saying that 3 apples plus 2 apples gives 5 apples.

合并同类项时，将系数相加或相减，字母部分保持不变。这类似于 3 个苹果加 2 个苹果等于 5 个苹果。

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

If the coefficients are negative, follow the same rule carefully. For example, −4y + 9y = 5y, because −4 + 9 = 5.

如果系数是负数，也要按照同样的规则仔细计算。例如，−4y + 9y = 5y，因为 −4 + 9 = 5。

When an expression has more than one type of variable, group the like terms together first.

当一个表达式含有多种变量时，应先把同类项分组。

5x + 3y − 2x + 7y = (5x − 2x) + (3y + 7y) = 3x + 10y

It is often helpful to underline or circle like terms with the same symbol before combining them.

在合并之前，用相同的符号将同类项划线或圈出通常很有帮助。

5. The Distributive Law: Expanding Brackets | 乘法分配律：去括号

When a number or letter is placed directly outside a bracket, every term inside the bracket is multiplied by that number or letter. This process is called expanding the brackets.

当一个数字或字母紧挨在括号外时，括号内的每一项都要乘以这个数字或字母。这个过程叫做去括号。

a(b + c) = ab + ac

For example, 3(x + 4) = 3x + 12, because 3 × x = 3x and 3 × 4 = 12.

例如，3(x + 4) = 3x + 12，因为 3 × x = 3x，3 × 4 = 12。

If there is a minus sign outside the bracket, the sign of every term inside the bracket must be reversed.

如果括号外是负号，括号内每一项的符号都必须变号。

−2(x − 5) = −2x + 10

Here, −2 multiplied by x gives −2x, and −2 multiplied by −5 gives +10. This is a very common place for errors, so take extra care.

这里，−2 乘以 x 得到 −2x，−2 乘以 −5 得到 +10。这是非常容易出错的地方，因此需要格外小心。

6. Index Laws in Multiplication | 乘法中的指数律

When simplifying expressions that contain powers, we use the index laws. The most important law for multiplication is that when the same base is multiplied, the powers are added.

在化简含有幂的表达式时，我们会使用指数律。乘法中最重要的指数律是：同底数的幂相乘，指数相加。

xᵃ × xᵇ = xᵃ⁺ᵇ

For example, x² × x³ = x⁵, because 2 + 3 = 5. This law applies only when the base letter is the same.

例如，x² × x³ = x⁵，因为 2 + 3 = 5。这条法则只在底数字母相同时适用。

When coefficients are present, multiply the coefficients separately and then apply the index law to the variable part.

当存在系数时，先把系数相乘，然后对字母部分应用指数律。

2x² × 3x³ = 6x⁵

The division law is also useful: when the same base is divided, subtract the powers.

除法法则也很有用：同底数的幂相除，指数相减。

xᵃ ÷ xᵇ = xᵃ⁻ᵇ

For example, 8x⁶ ÷ 2x² = 4x⁴, because 8 ÷ 2 = 4 and 6 − 2 = 4.

例如，8x⁶ ÷ 2x² = 4x⁴，因为 8 ÷ 2 = 4，6 − 2 = 4。

7. Simplifying Expressions with Brackets and Powers | 化简带有括号和幂的表达式

Many exam questions ask you to combine several skills: expand brackets, then collect like terms, and then apply index laws where necessary.

许多考试题目要求综合运用多种技能：先去括号，再合并同类项，必要时再应用指数律。

Work step by step and write each new line clearly. This reduces mistakes and shows the examiner how you reached your answer.

一步一步地计算，并清楚地写出每一步。这样可以减少错误，也能让阅卷人看到你的解题过程。

3(2x + 4) − 2(x − 1) = 6x + 12 − 2x + 2 = 4x + 14

In this example, first expand both brackets, then combine the x terms and the constant terms.

在这个例子中，先展开两个括号，然后合并 x 项和常数项。

If powers are included, multiply coefficients first, then add the indices for the same letters.

如果包含幂，先乘系数，再把相同字母的指数相加。

5x² × 2x³ − x⁴ = 10x⁵ − x⁴

Since 10x⁵ and −x⁴ have different powers, they are not like terms and cannot be combined further.

因为 10x⁵ 和 −x⁴ 的指数不同，它们不是同类项，不能继续合并。

8. Common Mistakes to Avoid | 常见错误与规避方法

One common mistake is adding terms that are not like terms. For example, 3x + 4y cannot be written as 7xy, because the letters are different.

一个常见错误是把不是同类项的项相加。例如，3x + 4y 不能写成 7xy，因为字母不同。

Another common mistake is forgetting to reverse the signs inside a bracket when the bracket has a negative sign in front of it.

另一个常见错误是，当括号前有负号时，忘记把括号内各项的符号变号。

−2(x − 5) = −2x + 10, not −2x − 10

Some students also confuse the addition law for indices with the multiplication law. Remember, x² + x² = 2x², but x² × x² = x⁴.

有些学生还会混淆指数的加法与乘法法则。记住，x² + x² = 2x²，而 x² × x² = x⁴。

Finally, do not divide by a variable unless you are sure it is not zero. At KS3 level, most simplification questions avoid this problem, but the idea is important.

最后，除非确定变量不为零，否则不要除以变量。在 KS3 阶段，大多数化简题会避免这种情况，但这个概念仍然重要。

9. Worked Example 1: Linear Expression | 例题一：线性表达式

Simplify the expression 7a + 4b − 3a + 9b − 2.

化简表达式 7a + 4b − 3a + 9b − 2。

First group the a terms, the b terms and the constant: (7a − 3a) + (4b + 9b) − 2.

首先将 a 项、b 项和常数项分组：(7a − 3a) + (4b + 9b) − 2。

Now combine the like terms: 7a − 3a = 4a, and 4b + 9b = 13b.

现在合并同类项：7a − 3a = 4a，4b + 9b = 13b。

7a + 4b − 3a + 9b − 2 = 4a + 13b − 2

The expression is now in its simplest form because there are no like terms left to combine.

这个表达式现在已经是最简形式，因为没有可以继续合并的同类项。

10. Worked Example 2: Expanding and Collecting Like Terms | 例题二：去括号并合并同类项

Simplify 4(2x + 3) + 5(x − 2).

化简 4(2x + 3) + 5(x − 2)。

First expand each bracket: 4 × 2x = 8x, 4 × 3 = 12, 5 × x = 5x, and 5 × −2 = −10.

首先展开每个括号：4 × 2x = 8x，4 × 3 = 12，5 × x = 5x，5 × −2 = −10。

Write the expanded expression: 8x + 12 + 5x − 10.

写出展开后的表达式：8x + 12 + 5x − 10。

Combine the x terms and the constants: 8x + 5x = 13x, and 12 − 10 = 2.

合并 x 项和常数项：8x + 5x = 13x，12 − 10 = 2。

4(2x + 3) + 5(x − 2) = 13x + 2

Check that the signs are correct after expanding. A quick numerical check with a simple value of x can help.

检查去括号后的符号是否正确。用一个简单的 x 值进行快速数值检验会有帮助。

11. Worked Example 3: Powers and Multiplication | 例题三：幂与乘法

Simplify 3x² × 4x⁵ − 2x⁴.

化简 3x² × 4x⁵ − 2x⁴。

Multiply the coefficients and add the powers of x in the first term: 3 × 4 = 12, and x² × x⁵ = x²⁺⁵ = x⁷.

将第一项中的系数相乘，并将 x 的指数相加：3 × 4 = 12，x² × x⁵ = x²⁺⁵ = x⁷。

So the first term becomes 12x⁷. The second term is −2x⁴.

因此第一项变为 12x⁷。第二项是 −2x⁴。

3x² × 4x⁵ − 2x⁴ = 12x⁷ − 2x⁴

The terms 12x⁷ and −2x⁴ are not like terms because the powers are different, so the expression cannot be simplified further.

12x⁷ 和 −2x⁴ 不是同类项，因为指数不同，所以这个表达式不能进一步化简。

12. Practice Questions and Answers | 练习题与答案

Try these questions before looking at the answers. Write every step and compare your method with the solutions.

在查看答案之前，先尝试这些题目。写出每一步，并将你的方法与答案进行对比。

Simplify 6m + 4n − 2m + 3n. 化简 6m + 4n − 2m + 3n。
Expand and simplify 5(2x + 3) − 3(x − 4). 展开并化简 5(2x + 3) − 3(x − 4)。
Simplify 2x³ × 5x². 化简 2x³ × 5x²。
Simplify 12y⁶ ÷ 3y². 化简 12y⁶ ÷ 3y²。

Answers: 4m + 7n; 7x + 27; 10x⁵; 4y⁴.

答案：4m + 7n；7x + 27；10x⁵；4y⁴。

If you made a mistake, identify whether it was in expanding brackets, combining like terms, or using the index laws. Focus your revision on that specific skill.

如果你做错了，要判断错误是在去括号、合并同类项，还是在运用指数律。将复习重点放在相应的技能上。

13. Key Takeaways and Exam Tips | 考点总结与考试技巧

Always remember that only like terms can be combined. When expanding brackets, multiply every term inside the bracket by the term outside, and be careful with negative signs.

始终记住：只有同类项才能合并。去括号时，括号内的每一项都要乘以括号外的项，并且要特别小心负号。

For powers, multiply coefficients and add indices when the base is the same. For division, subtract the indices. Do not apply these rules to addition or subtraction.

对于幂，当底数相同时，系数相乘，指数相加。对于除法，指数相减。不要把这些法则用于加法或减法。

Finally, always present your answer in its simplest form. Check each line for sign errors and make sure no like terms remain.

最后，始终以最简形式呈现答案。逐行检查符号错误，确保没有同类项遗漏。

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Simplifying Algebraic Expressions | 代数式化简