Edexcel IGCSE Maths: Multiplication & Division of Algebraic Fractions | Edexcel IGCSE数学：代数分式的乘除运算

📚 Edexcel IGCSE Maths: Multiplication & Division of Algebraic Fractions | Edexcel IGCSE数学：代数分式的乘除运算

Algebraic fractions are a core topic in the Edexcel IGCSE Mathematics syllabus (4MA1). Questions on multiplying and dividing algebraic fractions appear frequently in Paper 1F, Paper 1H, Paper 2F and Paper 2H, and they test your ability to combine factorisation with the same fraction rules you already know from arithmetic.

代数分式是 Edexcel IGCSE 数学（4MA1）考纲的核心内容。乘除运算的题目在 Paper 1F、1H、2F 与 2H 中频繁出现，考查的是你将因式分解与算术中已有的分数运算法则结合使用的能力。

1. The Rules You Must Know | 必须掌握的运算法则

Before touching algebra, recall the rules for numeric fractions. To multiply two fractions, multiply the numerators and multiply the denominators: a/b × c/d = ac/bd.

接触代数之前，先复习数字分数的运算法则。两个分数相乘，就是分子乘分子、分母乘分母：a/b × c/d = ac/bd。

a/b × c/d = ac/bd

To divide by a fraction, multiply by its reciprocal (the fraction flipped upside down): a/b ÷ c/d = a/b × d/c.

除以一个分数，等于乘上它的倒数（即把分数上下颠倒）：a/b ÷ c/d = a/b × d/c。

a/b ÷ c/d = a/b × d/c

These two rules work exactly the same way when a, b, c and d are algebraic expressions. For example, 2x/3 × 5/x = 10x/(3x) = 10/3.

当 a、b、c、d 是代数式时，这两条法则完全同样适用。例如，2x/3 × 5/x = 10x/(3x) = 10/3。

2. Golden Rule: Factorise Before You Operate | 黄金法则：运算之前先因式分解

This is the single most important habit for algebraic fractions. Factorise every numerator and every denominator completely before multiplying or dividing. This exposes the common factors which can then be cancelled.

这是处理代数分式最重要的一个习惯。在相乘或相除之前，先把每一个分子和分母都彻底因式分解。这样才能暴露出可以约掉的公因式。

For example, x² – 9 is not obviously related to x + 3, but once you write x² – 9 = (x – 3)(x + 3), you can connect the two expressions and cancel.

例如，x² – 9 看起来与 x + 3 没有直接关系，但一旦把 x² – 9 写成 (x – 3)(x + 3)，两者就能联系起来并约分。

The three factorisation techniques you will need most are: taking out a common factor (e.g. 6x² + 3x = 3x(2x + 1)); difference of two squares (x² – a² = (x – a)(x + a)); and factorising quadratics (x² + (a + b)x + ab = (x + a)(x + b)).

你最常用的三种因式分解技巧是：提取公因式（如 6x² + 3x = 3x(2x + 1)）；平方差公式（x² – a² = (x – a)(x + a)）；以及二次三项式分解（x² + (a + b)x + ab = (x + a)(x + b)）。

3. Multiplying Algebraic Fractions | 代数分式的乘法

Step 1: factorise each numerator and denominator. Step 2: multiply the remaining numerators together and the remaining denominators together. Step 3: cancel any factors that appear in both the numerator and the denominator.

第一步：分解每个分子和分母。第二步：把剩下的分子相乘、分母相乘。第三步：约去分子分母中共同的因式。

Worked example: simplify (x² – 4)/(x² + 2x) × x/(x – 2).

例题

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