Simplifying Algebraic Fractions | 代数分式的化简

📚 Simplifying Algebraic Fractions | 代数分式的化简

Algebraic fractions are fractions in which the numerator and/or the denominator contain algebraic expressions, such as x² or x + 1. In the Edexcel IGCSE Mathematics syllabus, the ability to simplify algebraic fractions accurately is essential for solving equations, sketching graphs and working with functions. Mastering this topic will allow you to handle even the most complicated-looking fractions with confidence.

代数分式是指分子和（或）分母中含有代数表达式（如 x² 或 x + 1）的分数。在 Edexcel IGCSE 数学大纲中，准确化简代数分式的能力对于解方程、绘制函数图像以及处理函数问题都至关重要。掌握本专题后，你就能自信地应对各种看似复杂的代数分式。

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

An algebraic fraction takes the form P/Q, where P and Q are algebraic expressions and Q ≠ 0. Examples include 3/x, (x + 1)/(x – 2) and (x² – 4)/(x² + 3x + 2).

代数分式的形式为 P/Q，其中 P 和 Q 是代数表达式且 Q ≠ 0。例如：3/x、(x + 1)/(x – 2) 和 (x² – 4)/(x² + 3x + 2)。

Just like numerical fractions, algebraic fractions can be simplified, added, subtracted, multiplied and divided. However, before performing any operation, you should always factorise all numerators and denominators completely.

与数值分数一样，代数分式也可以进行化简、加减、乘除运算。但在进行任何运算之前，你应当先将所有分子和分母完全因式分解。

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

Before you can cancel, multiply or divide, you must express every quadratic expression as a product of linear factors. The three factorisation techniques you will need are:

在约分、乘除之前，你必须将每个二次表达式写成线性因式的乘积。你需要掌握的三种因式分解技巧是：

• Taking out a common factor: 2x² + 6x = 2x(x + 3)

• 提取公因式：2x² + 6x = 2x(x + 3)

• Difference of two squares: x² – 9 = (x + 3)(x – 3)

• 平方差公式：x² – 9 = (x + 3)(x – 3)

• Quadratic factorisation: x² – x – 6 = (x – 3)(x + 2)

• 二次三项式因式分解：x² – x – 6 = (x – 3)(x + 2)

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