Using Partial Fractions for Edexcel A-Level Maths | Edexcel A-Level 数学：部分分式的应用

📚 Using Partial Fractions for Edexcel A-Level Maths | Edexcel A-Level 数学：部分分式的应用

Partial fractions are a powerful algebraic tool in A-Level Mathematics. They allow you to break a complicated rational function into a sum of simpler fractions, making integration, binomial expansion and curve analysis far easier. In the Edexcel specification, this skill is closely linked to Pure Mathematics topics such as algebra, functions, calculus and series.

部分分式是 A-Level 数学中一个强大的代数工具。它可以把复杂的有理函数拆成几个较简单分式之和，从而使积分、二项展开和曲线分析变得更加容易。在 Edexcel 考试大纲中，这一技能与纯数学中的代数、函数、微积分和级数等主题紧密相关。

1. The Core Idea of Partial Fractions | 部分分式的核心思想

A rational function is the ratio of two polynomials. If the degree of the numerator is less than the degree of the denominator, the fraction is called proper. The goal of partial fractions is to express a proper rational function as a sum of simpler ‘partial fractions’ whose denominators are the factors of the original denominator.

有理函数是两个多项式之比。如果分子的次数低于分母的次数，这个分式就称为真分式。部分分式的目标，就是把一个真有理函数表示为若干个较简单的“部分分式”之和，而这些部分分式的分母正是原分母的因式。

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

For example, the expression above can be rewritten with constants A and B. This decomposition is unique as long as the original fraction is proper.

例如，上面的表达式可以改写为带有常数 A 和 B 的形式。只要原分式是真分式，这种分解就是唯一的。

This idea is not just an algebraic trick; it provides a systematic way to handle inverse processes such as integration and series expansion.

这一思想并不只是一个代数技巧；它为积分和级数展开等逆过程提供了一种系统化的处理方法。

2. Rational Functions and Proper Fractions | 有理函数与真分式

Before decomposing, always check whether the rational function is proper. If the degree of the numerator is greater than or equal to the degree of the denominator, it is improper and must be simplified first.

在分解之前，一定要先判断有理函数是否为真分式。如果分子的次数大于或等于分母的次数，它就是假分式，必须先进行化简。

In an improper fraction, division of polynomials gives a quotient polynomial plus a proper remainder. This remainder is then the part that can be decomposed into partial fractions.

对于假分式，通过多项式除法可以得到一个商多项式加上一个真余式。这个真余式才是可以继续分解为部分分式的部分。

Edexcel exam questions often hide this step inside integration or expansion problems, so recognising an improper fraction early saves time and avoids invalid decompositions.

Edexcel 考试题目经常把这一步隐藏在积分或展开问题中，因此尽早识别出假分式可以节省时间，并避免无效的分解。

3. Polynomial Division First | 先进行多项式除法

Suppose you need to express (x³ + 2x² + 5) / ((x – 1)(x + 2)) in partial fractions. The numerator degree is 3 and the denominator degree is 2, so the fraction is improper. Use long division or algebraic division to write it as a polynomial plus a proper fraction.

假设你需要把 (x³ + 2x² + 5) / ((x – 1)(x + 2)) 表示成部分分式。分子的次数是 3，分母的次数是 2，因此这是假分式。使用长除法或代数除法，把它写成一个多项式加上一个真分式。

Original fraction = Q(x) + R(x) / D(x), where deg R(x) < deg D(x)

In general, after division you obtain the form shown above. Then only R(x) / D(x) is decomposed into partial fractions.

一般来说，除法后你会得到上面这种形式。然后只需对 R(x) / D(x) 进行部分分式分解。

Writing the quotient correctly is important because it contributes polynomial terms to the final answer, especially when integrating or expanding term-by-term.

正确写出商非常重要

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