Algebraic Fractions | 代数分式

📚 Algebraic Fractions | 代数分式

Algebraic fractions are rational expressions that often appear in Edexcel A-Level Pure Mathematics. You will need to simplify them, combine them, and solve equations in which they appear.

代数分式是 Edexcel A-Level 纯数学中经常出现的有理表达式。你需要掌握化简、加减乘除以及解含有代数分式的方程。

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

An algebraic fraction is a fraction whose numerator and denominator are polynomials. For example, (x² + 3x + 2)/(x + 1) and 2/(x² – 4) are algebraic fractions.

代数分式是分子和分母都为多项式的分式。例如 (x² + 3x + 2)/(x + 1) 和 2/(x² – 4) 都是代数分式。

In Edexcel exams, questions usually ask you to simplify, add, subtract, multiply, divide, or solve equations involving such expressions.

在 Edexcel 考试中，题目通常要求你对这类表达式进行化简、加减乘除运算，或求解相关方程。

2. Simplifying by Factorising | 通过因式分解化简

The first step is to factorise both numerator and denominator completely. Then cancel common factors, but only if they are multiplied by the whole numerator and denominator.

第一步是把分子和分母完全因式分解，然后约去公因式。但只有当公因式是以乘法形式作用于整个分子和分母时才能约分。

(x² – 9)/(x² – x – 12) = (x – 3)(x + 3)/[(x – 4)(x + 3)] = (x – 3)/(x – 4)

例如，上式通过平方差和十字相乘分解后，约去 (x + 3) 得到 (x – 3)/(x – 4)。

You cannot cancel terms across a plus or minus sign. For instance, in (x + 3)/(x – 4), x is not cancelled because x is a term, not a factor.

你不能约去加减号连接的项。例如在 (x + 3)/(x – 4) 中，x 不能被约去，因为 x 是项而不是因式。

3. Domain Restrictions | 定义域限制

Before cancelling, identify values that make the original denominator zero. These values are excluded from the domain.

在约分之前，要找出使原分母为零的值，这些值必须从定义域中排除。

For (x² – 9)/(x² – x – 12), the denominator is zero when (x – 4)(x + 3) = 0, so x ≠ 4 and x ≠ -3.

对于 (x² – 9)/(x² – x – 12)，分母在 (x – 4)(x + 3) = 0 时为零，因此 x ≠ 4 且 x ≠ -3。

Even though (x + 3) is cancelled, x = -3 is still not in the domain of the original expression. In simplified form, write x ≠ 4, x ≠ -3.

即使 (x + 3) 被约去，x = -3 仍然不在原表达式的定义域中。化简后应注明 x ≠ 4, x ≠ -3。

4. Multiplying and Dividing | 乘法与除法

To multiply algebraic fractions, factorise all numerators and denominators, cancel any common factors across the product, then multiply remaining factors.

代数分式相乘时，先将所有分子分母因式分解，在乘积中约去公因式，然后相乘剩余因式。

(x² – 1)/(x + 2) × (x + 2)/(x – 1) = (x – 1)(x + 1)(x + 2)/[(x + 2)(x – 1)] = x + 1

例如，约去 (x – 1) 和 (x + 2) 后，结果为 x + 1，但需注明 x ≠ -2, x ≠ 1。

To divide, multiply by the reciprocal of the second fraction, then follow the same factor-cancel method.

除法运算时，将第二个分式取倒数后相乘，然后按相同方法因式

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