Topic 115: Algebraic Fractions | 专题115：代数分式

📚 Topic 115: Algebraic Fractions | 专题115：代数分式

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. They follow the same rules as numerical fractions, but require careful handling of variables and factorisation. In this revision guide, we will cover the essential skills you need for the Edexcel IGCSE Mathematics exam.

代数分式是指分子、分母或两者都含有代数表达式的分数。它们遵循与数值分数相同的运算法则，但需要谨慎处理变量和因式分解。在本复习指南中，我们将涵盖Edexcel IGCSE数学考试所需的关键技能。

1. Simplifying Algebraic Fractions | 化简代数分式

To simplify an algebraic fraction, factorise the numerator and the denominator completely, then cancel the common factors. This is only valid when the factor is not equal to zero.

化简代数分式时，需要先将分子和分母完全因式分解，然后约去公因式。只有当该因式不等于零时，约分才成立。

Factorise both top and bottom separately.
Cancel common linear or quadratic factors.
State restrictions if the cancelled factor could make the denominator zero.

分别对分子和分母进行因式分解。
约去共同的一次或二次因式。
如果约去的因式可能使分母为零，需注明限制条件。

Example: Simplify (x² – 9) ⁄ (x² – 3x)

x² – 9 = (x – 3)(x + 3), x² – 3x = x(x – 3)

So the expression = (x – 3)(x + 3) ⁄ [x(x – 3)] = (x + 3) ⁄ x, x ≠ 0, x ≠ 3

Always check if a quadratic can be factorised using the difference of two squares or simple trinomial patterns.

始终检查二次式是否能用平方差公式或简单的三项式模式进行因式分解。

2. Adding and Subtracting Algebraic Fractions | 代数分式的加法与减法

To add or subtract algebraic fractions, you need a common denominator. The most efficient way is to use the lowest common denominator (LCD) of the given denominators.

进行加减运算时，需要先求出公分母。最有效的方法是找出给定分母的最简公分母（LCD）。

Find the LCD of all denominators.
Rewrite each fraction with the LCD as its new denominator.
Add or subtract the numerators, then simplify the result.

找出所有分母的最简公分母。
将每个分数改写为以最简公分母为分母的形式。
对分子进行加减，然后化简结果。

Example: 2 ⁄ (x+1) + 3 ⁄ (x–2)

LCD = (x+1)(x–2)

= [2(x–2) + 3(x+1)] ⁄ [(x+1)(x–2)]

= (2x – 4 + 3x + 3) ⁄ [(x+1)(x–2)]

= (5x – 1) ⁄ [(x+1)(x–2)]

Be careful with subtraction: place the entire second numerator in brackets before simplifying.

做减法时需小心：先将第二个分子整体放入括号，再去括号化简。

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

Multiplying algebraic fractions is straightforward: multiply the numerators together and multiply the denominators together. Factorise before multiplying to simplify the process.

代数分式的乘法很简单：分子乘分子，分母乘分母。最好先因式分解再进行乘法运算，以便约分。

Factorise all numerators and denominators completely.
Cancel any common factors across the top and bottom.
Multiply the remaining terms.

将所有分子和分母完全因式分解。
上下约去所有公因式。
将剩余项相乘。

Example: (x² – 1) ⁄ (x+2) × (x+2) ⁄ (x–1)

= [(x–1)(x+1)] ⁄ (x+2) × (x+2) ⁄ (x–1)

= x+1, x ≠ –2, x ≠ 1

Notice that we cancel the (x+2) and (x–1) terms before multiplying, making the result much simpler.

注意我们在相乘前先约去了(x+2)和(x–1)，使结果更加简洁。

4. Dividing Algebraic Fractions | 代数分式的除法

Division of algebraic fractions is performed by multiplying by the reciprocal of the second fraction. Remember to flip the second fraction (the divisor) and change the division sign to multiplication.

代数分式的除法通过乘以第二个分数的倒数来完成。记得将第二个分数（除数）颠倒，并把除号改为乘号。

Keep the first fraction unchanged.
Change division to multiplication.
Flip the second fraction (find its reciprocal).
Factorise and simplify where possible.

保持第一个分数不变。
将除号改为乘号。
颠倒第二个分数（求其倒数）。
尽可能因式分解并化简。

Example: (x² – 4) ⁄ 5 ÷ (x – 2) ⁄ 15

= (x² – 4) ⁄ 5 × 15 ⁄ (x – 2)

= [(x–2)(x+2)] ⁄ 5 × 15 ⁄ (x–2)

= 3(x+2), x ≠ 2

Always state the excluded values for which any denominator would be zero.

始终注明使任何分母为零的排除值。

5. Solving Equations with Algebraic Fractions | 解含代数分式的方程

To solve an equation containing algebraic fractions, multiply both sides of the equation by the common denominator to eliminate the fractions. Then solve the resulting linear or quadratic equation.

解含代数分式的方程时，需在方程两边同时乘以公分母以消去分数，然后解所得的线性或二次方程。

Identify the least common denominator (LCD) of all fractions.
Multiply every term on both sides by the LCD.
Simplify and solve the resulting equation.
Check your answers by substituting back into the original equation.

找出所有分式的最简公分母（LCD）。
将方程两边的每一项都乘以LCD。
化简并解所得方程。
将解代回原方程进行检验。

Example: 1 ⁄ (x+1) + 2 ⁄ (x–1) = 3

Multiply by (x+1)(x–1):

(x–1) + 2(x+1) = 3(x+1)(x–1)

x – 1 + 2x + 2 = 3(x² – 1)

3x + 1 = 3x² – 3

3x² – 3x – 4 = 0

This quadratic may require the quadratic formula or completing the square to solve.

这个二次方程可能需要使用求根公式或配方法来求解。

6. Working with Mixed Algebraic Expressions | 处理混合代数表达式

Sometimes an algebraic expression contains a whole number term and a fraction. When solving or simplifying, treat the whole number as a fraction with denominator 1.

有时代数表达式会包含整数项和分数项。在化简或求解时，将整数视为分母为1的分数。

Write the integer over 1.
Combine with the other fractions using a common denominator.
Simplify carefully, especially when negative signs are involved.

将整数写为分母为1的形式。
使用公分母与其他分数合并。
特别注意负号影响，细致化简。

Example: x + 3 ⁄ (x–2) = (x+1) ⁄ 2

Multiply all terms by 2(x–2):

2x(x–2) + 6 = (x+1)(x–2)

Expanding and collecting terms gives a quadratic equation that may have two solutions.

展开并整理后得到二次方程，通常有两个解。

7. Algebraic Fractions and Quadratic Factorisation | 代数分式与二次因式分解

Many algebraic fraction problems require you to recognise standard factorisations: difference of two squares, perfect squares, and simple trinomials.

许多代数分式问题需要识别标准因式分解形式：平方差、完全平方以及简单三项式。

a² – b² = (a + b)(a – b)
a² + 2ab + b² = (a + b)²
a² – 2ab + b² = (a – b)²
x² + (p+q)x + pq = (x+p)(x+q)

a² – b² = (a + b)(a – b)
a² + 2ab + b² = (a + b)²
a² – 2ab + b² = (a – b)²
x² + (p+q)x + pq = (x+p)(x+q)

When a cubic or higher-degree factor appears, try to factor out a common term first.

若出现三次或更高次因式，先尝试提取公因式。

8. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Students often lose marks on algebraic fractions due to simple errors. Here are the most common pitfalls and how to avoid them.

学生在代数分式上常因简单错误丢分。以下是最常见的陷阱及避免方法。

Mistake	Correction	Example
Cancelling terms that are not factors	Only cancel common factors (multiplied terms), never isolated terms	(x+2)⁄(x+5) cannot be simplified
Forgetting to multiply all terms by the LCD	Multiply every term on both sides of the equation	x + 1⁄x = 2 → multiply by x: x² + 1 = 2x
Losing negative signs during subtraction	Put the numerator in brackets before subtracting	1⁄(x) – (x–1)⁄(x) = (1–x+1)⁄x = (2–x)⁄x
Not stating when the denominator equals zero	Always exclude values that make any denominator zero	For 1⁄(x–2), state x ≠ 2

Review each line carefully and test your final answer by substitution.

仔细检查每一步，并通过代值验证最终答案。

9. Worked Exam-Style Examples | 考试风格例题精讲

Let us work through two full problems that reflect typical Edexcel IGCSE questions.

我们完整解答两道具有典型Edexcel IGCSE风格的题目。

Problem 1: Simplify (3 – x) ⁄ (x² – 9)

Rewrite 3 – x as –(x – 3). Then x² – 9 = (x – 3)(x + 3).

将3 – x改写为–(x – 3)。又x² – 9 = (x – 3)(x + 3)。

= –(x – 3) ⁄ [(x – 3)(x + 3)] = –1 ⁄ (x + 3), x ≠ 3

Problem 2: Solve 2 ⁄ (x+1) – 1 ⁄ (x–1) = 1 ⁄ 3

LCD = 3(x+1)(x–1). Multiply every term by the LCD:

最简公分母为3(x+1)(x–1)。将每项乘以LCD：

6(x–1) – 3(x+1) = (x+1)(x–1)

6x – 6 – 3x – 3 = x² – 1

3x – 9 = x² – 1

x² – 3x + 8 = 0

The discriminant Δ = (–3)² – 4×1×8 = 9 – 32 = –23 < 0, so there is no real solution.

判别式Δ = (–3)² – 4×1×8 = 9 – 32 = –23 < 0，因此没有实数解。

10. Practice Questions | 练习与巩固

Test yourself with these five questions. Try to complete them without your notes first.

用这五道题自测。先尝试不看笔记完成。

Simplify: (2x² – 8) ⁄ (3x – 6)
Simplify: (x² + 5x + 6) ⁄ (x² + 3x) × (x ⁄ (x+2))
Solve: 1 ⁄ (x) + 1 ⁄ (x+2) = 3 ⁄ 4
Simplify: (x² – 1) ⁄ (x² – 4x + 3) ÷ (x+1) ⁄ (x–3)
Solve: (x+2) ⁄ (x–2) – (x–1) ⁄ (x+1) = 5 ⁄ 8

化简：(2x² – 8) ⁄ (3x – 6)
化简：(x² + 5x + 6) ⁄ (x² + 3x) × (x ⁄ (x+2))
解方程：1 ⁄ x + 1 ⁄ (x+2) = 3 ⁄ 4
化简：(x² – 1) ⁄ (x² – 4x + 3) ÷ (x+1) ⁄ (x–3)
解方程：(x+2) ⁄ (x–2) – (x–1) ⁄ (x+1) = 5 ⁄ 8

Answers: 1. 2(x+2)⁄3 (x≠2) 2. x⁄(x+3) (x≠0, –2, –3) 3. x = 2 or x = –4⁄3 4. (x–1)⁄(x–3) (x≠3, –1, 1) 5. x = 5 or x = –3⁄5

答案：1. 2(x+2)⁄3 (x≠2) 2. x⁄(x+3) (x≠0, –2, –3) 3. x = 2 或 x = –4⁄3 4. (x–1)⁄(x–3) (x≠3, –1, 1) 5. x = 5 或 x = –3⁄5

11. Exam Tips for Algebraic Fractions | 代数分式考试技巧

In the IGCSE exam, algebraic fraction questions often appear in both Paper 1 (non-calculator) and Paper 2 (calculator allowed). Use these tips to maximise your marks.

在IGCSE考试中，代数分式相关题目常在Paper 1（不可用计算器）和Paper 2（可用计算器）中出现。运用以下技巧可最大化得分。

Always factorise before simplifying or solving.
Write down every step; partial marks are usually awarded.
When solving equations, check for extraneous roots that make a denominator zero.
If a quadratic factor does not cancel, try rearranging or multiplying out carefully.
Use the quadratic formula when factorisation is not obvious.

在化简或求解之前始终先因式分解。
逐步书写；通常步骤分是按步给分。
解方程时，要检查使分母为零的增根。
如果二次因式不能约去，尝试重新整理或仔细展开。
当因式分解不明显时，使用求根公式。

Memorise standard factorisation patterns early, and practise past paper questions involving fractions.

尽早熟记标准因式分解模式，并通过历年真题练习涉及分数的题目。

12. Summary | 小结

Algebraic fractions are a core topic in Edexcel IGCSE Mathematics. Mastering them requires confidence in factorisation, manipulation of equations, and careful attention to domain restrictions. Regular practice will make the process automatic.

代数分式是Edexcel IGCSE数学的核心考点。掌握它们需要具备坚定的因式分解能力、方程操作技巧，并密切关注定义域限制。经常练习将使解题过程变得自动化。

Simplify by factorising and cancelling common factors.
Add and subtract using the lowest common denominator.
Multiply across, divide by flipping the second fraction.
Solve equations by multiplying through by the LCD.
Always exclude values that make any denominator zero.

通过因式分解和约去公因式进行化简。
使用最简公分母进行加减运算。
乘法直接相乘，除法颠倒第二个分数后相乘。
解方程时两边乘以公分母以去分母。
始终排除使任何分母为零的值。

Keep this guide handy, work through the practice questions, and you will be well prepared for whatever fraction problem the exam throws at you.

将本指南留存备用，完成练习题目，你就能从容应对考试中的任何分式问题。

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Topic 115: Algebraic Fractions | 专题115：代数分式