Simplifying Algebraic Fractions | 简化代数分式

📚 Simplifying Algebraic Fractions | 简化代数分式

Algebraic fractions appear throughout IGCSE Edexcel mathematics. To “simplify” an algebraic fraction, we rewrite it in the simplest equal form, usually by factorising the numerator and denominator and cancelling common factors. This article explains the rules, works through multiple examples, and highlights the mistakes that lose marks.

代数分式在 IGCSE Edexcel 数学中随处可见。“化简”代数分式，就是把它改写成与之相等的最简形式，通常通过把分子和分母因式分解并约去公因式来完成。本文会解释运算法则、演示多个例题，并帮你避开最常见的失分点。

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1. What Are Algebraic Fractions? | 什么是代数分式？

An algebraic fraction is a fraction in which the numerator or the denominator contains a variable expression. In IGCSE Edexcel mathematics, you will meet fractions such as 3/x, (x+5)/(x−2), and (x² − 4x + 3)/(x² − 1).

代数分式是指分子或分母中含有变量表达式的分数。在 IGCSE Edexcel 数学中，你会遇到如 3/x、(x+5)/(x−2) 和 (x² − 4x + 3)/(x² − 1) 这样的分式。

The word “simplify” means to write the fraction in its simplest equivalent form. This usually involves factorising all algebraic parts and then cancelling common factors, exactly as you would do with numerical fractions such as 6/8 = 3/4.

“化简”的意思是把这个分式写成与其等价的最简形式。通常需要先对所有代数部分进行因式分解，再约去公因式，就像化简数字分数 6/8 = 3/4 一样。

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2. Factorising First: The Golden Rule | 先因式分解：黄金法则

The single most important rule is: factorise before you cancel. A fraction can only be simplified when the numerator and denominator contain a common factor, not merely a common term. For example, in the fraction (x+3)/(x+5), we cannot cancel the x’s because x is part of a sum, not a factor. However, (3x)/(5x) can be simplified to 3/5 because x is a multiplier of both the numerator and denominator.

最重要的一条法则是：先因式分解，再约分。只有分子和分母含有相同的“因式”时，才能化简；如果只是含有相同的“项”，比如 (x+3)/(x+5)，绝不能把两个 x 约掉，因为这里的 x 是相加项而不是因子。而 (3x)/(5x) 可以化简为 3/5，因为 x 同时是两者的因数。

You should be confident with these basic factorising forms:

你需要熟练掌握以下基本因式分解形式：

Expression	Factorised form
2x² + 6x	2x(x + 3)
x² − 7x + 12	(x − 3)(x − 4)
x² − 16	(x + 4)(x − 4)

Always look for a common factor first. For example, 3x² − 12 should be written as 3(x² − 4), and then as 3(x − 2)(x + 2).

永远先找公因式。例如，3x² − 12 应先写成 3(x² − 4)，再进一步写成 3(x − 2)(x + 2)。

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3. Cancelling Common Factors: The Basic Rule | 约去公因式：基本法则

For any algebraic fraction of the form P·A / P·B, where P is a non-zero common factor, we may cancel P to obtain A/B.

对于任何形如 P·A / P·B 的代数分式，如果 P 是非零公因式，我们就可以约去 P，得到 A/B。

(P·A)/(P·B) = A/B

For example:

例如：

(3x²)/(6x) = (3·x·x)/(3·2·x) = x/2

Here we cancelled the common factor 3x because 3x divides evenly into both the numerator and denominator. Strictly speaking, the cancellation is valid only where x ≠ 0.

这里我们约去了公因式 3x，因为 3x 能同时整除分子和分母。严格来说，这个约分只在 x ≠ 0 时才成立。

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4. Worked Example with Quadratics | 二次多项式例题

Consider the fraction:

考虑以下分式：

(x² + 5x + 6)/(x² + 2x − 3)

Step 1: Factorise the numerator.

第一步：因式分解分子。

x² + 5x + 6 = (x + 2)(x + 3)

Step 2: Factorise the denominator.

第二步：因式分解分母。

x² + 2x − 3 = (x − 1)(x + 3)

Step 3: Cancel the common factor (x + 3).

第三步：约去公因式 (x + 3)。

((x + 2)(x + 3))/((x − 1)(x + 3)) = (x + 2)/(x − 1)

The simplified fraction is (x + 2)/(x − 1), provided x ≠ −3 and x ≠ 1.

化简结果为 (x + 2)/(x − 1)，其中需要满足 x ≠ −3 且 x ≠ 1。

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5. Difference of Two Squares | 平方差公式

The difference of two squares is one of the most useful factorising patterns in algebraic fractions:

平方差公式是代数分式中最常用的因式分解模式之一：

a² − b² = (a + b)(a − b)

Example 1:

例 1：

(x² − 9)/(x² + 3x) = ((x − 3)(x + 3))/(x(x + 3)) = (x − 3)/x

Here the common factor x + 3 was cancelled. The restriction is x ≠ 0 and x ≠

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