Algebraic Fractions: Simplifying, Adding, and Solving | 代数分式：化简、加减与求解

📚 Algebraic Fractions: Simplifying, Adding, and Solving | 代数分式：化简、加减与求解

Algebraic fractions appear in many IGCSE Mathematics questions. They follow the same rules as numerical fractions, but they also require careful factorisation and attention to values that make the denominator zero.

代数分式在 IGCSE 数学考试中很常见。它遵循与普通分数相同的运算法则，但还需要我们熟练地进行因式分解，并注意使分母等于零的取值。

1. What Is an Algebraic Fraction? | 什么是代数分式？

An algebraic fraction is a fraction whose numerator or denominator contains a variable. Examples include 3/(x+2) and (x²-1)/(x-3).

代数分式是指分子或分母中含有变量的分数。例如：3/(x+2) 和 (x²-1)/(x-3)。

Important: The denominator cannot be zero, so we must state restrictions such as x ≠ -2 or x ≠ 3.
注意：分母不能为零，因此我们必须注明限制条件，例如 x ≠ -2 或 x ≠ 3。

2. Simplifying Algebraic Fractions | 化简代数分式

To simplify an algebraic fraction, factorise the numerator and denominator completely, then cancel any common factors.

化简代数分式时，先把分子和分母完全因式分解，再约去所有公因式。

(x² – 4)/(x² + 2x) = ((x – 2)(x + 2))/(x(x + 2)) = (x – 2)/x

Here we cancel the factor (x + 2), not any single term. The restrictions are x ≠ 0 and x ≠ -2.

这里我们约去公因式 (x + 2)，而不是单独约去某项。限制条件为 x ≠ 0 且 x ≠ -2。

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

Multiply numerators together and denominators together. It is usually easier to cancel common factors before multiplying.

分子相乘、分母相乘。通常先约去公因式会让计算更简单。

(2x/(x + 3)) × ((x + 3)/x²) = 2/x

We cancel (x + 3) and one factor of x. The restrictions are x ≠ 0 and x ≠ -3.

我们约去了 (x + 3) 和 x 的一个幂次。限制条件为 x ≠ 0 且 x ≠ -3。

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

To divide by a fraction, multiply by its reciprocal, then simplify as usual.

除以一个分式等于乘以它的倒数，然后再化简。

(x + 1)/(x – 2) ÷ (x + 1)/(x² – 4) = (x + 1)/(x – 2) × ((x – 2)(x + 2))/(x + 1) = x + 2

Remember that x ≠ -1, x ≠ 2, and x ≠ -2 based on the original denominators.

根据原分母可知 x ≠ -1、x ≠ 2 且 x ≠ -2。

5. Adding and Subtracting with the Same Denominator | 同分母代数分式的加减

When the denominators are the same, simply add or subtract the numerators and keep the denominator unchanged.

当分母相同时，直接把分子相加或相减，分母保持不变。

2/(x + 1) + 3/(x + 1) = 5/(x + 1)

The restriction is x ≠ -1 because the denominator would otherwise be zero.

限制条件为 x ≠ -1，因为此时分母会变为零。

6. Adding and Subtracting with Different Denominators | 异分母代数分式的加减

First find the lowest common denominator (LCD), rewrite each fraction, then combine the numerators.

先找到最简公分母 (LCD)，改写每个分式，再合并分子。

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

The restrictions are x ≠ 0 and x ≠ -1.

限制条件为 x ≠ 0 且 x ≠ -1。

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

To solve an equation involving algebraic fractions, multiply every term by the common denominator, then solve the resulting linear or quadratic equation.

解含代数分式的方程时，将方程两边每一项都乘以公分母，然后解得到的整式方程。

Step 1: Factorise denominators if possible.
步骤一：如果可以，先对分母进行因式分解。
Step 2: Identify the common denominator and multiply both sides.
步骤二：确定公分母，并把方程两边都乘以它。
Step 3: Solve the resulting equation.
步骤三：解化简后的方程。
Step 4: Check that no solution makes an original denominator zero.
步骤四：检查是否有解使原分母为零，如果有应舍去。

8. Worked Exam-Style Example | 考试典型例题

Solve the equation 2/(x + 1) + 1/(x – 1) = 1.

解方程 2/(x + 1) + 1/(x – 1) = 1。

First, the common denominator is (x + 1)(x – 1). Multiply every term by it:

首先公分母为 (x + 1)(x – 1)。将每一项都乘以公分母：

2(x – 1) + 1(x + 1) = (x + 1)(x – 1)

Expand and simplify:

展开并化简：

3x – 1 = x² – 1

x² – 3x = 0

x(x – 3) = 0

So x = 0 or x = 3. Neither value makes the original denominators zero, so both are valid.

因此 x = 0 或 x = 3。这两个值都不会让原分母变为零，所以都有效。

9. Common Mistakes to Avoid | 常见错误提醒

Cancelling terms instead of factors: You can only cancel common factors, not individual terms.
错把“项”当“因式”约分：只能约去公因式，不能约去单独的项。
Forgetting restrictions: Always state values of x that make a denominator zero.
忘记限制条件：必须注明使分母为零的 x 值。
Sign errors in subtraction: When subtracting a numerator, use brackets to avoid sign mistakes.
减法中符号错误：进行减法时，建议给分子加上括号，防止符号出错。
Not checking solutions: After solving, verify that each solution is allowed.
不检验解：解完方程后，要确认每个解都满足原方程的限制条件。

10. Quick Revision Checklist | 快速复习清单

Factorise before simplifying.
化简前先因式分解。
Find the LCD before adding or subtracting.
加减前先找到最简公分母。
Multiply by the reciprocal when dividing.
做除法时乘以倒数。
State restrictions whenever a denominator is present.
只要分母中含有变量，就要注明限制条件。
Check answers after solving equations.
解方程后一定要检验答案。

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Algebraic Fractions: Simplifying, Adding, and Solving | 代数分式：化简、加减与求解