Partial Fractions | 部分分式

📚 Partial Fractions | 部分分式

Partial fractions are an algebraic technique used to break a complicated rational expression into a sum of simpler fractions. This makes integration, binomial expansion, and the analysis of rational functions much easier. In Edexcel A Level Mathematics, partial fractions appear regularly in Pure Mathematics, especially within integration and series work.

部分分式是一种代数技巧，用于把一个复杂的有理式拆分成若干简单分式之和。这使得积分、二项展开以及有理函数的分析变得容易得多。在 Edexcel A Level 数学中，部分分式经常在纯数学中出现，尤其是在积分和级数部分。

1. What Are Partial Fractions? | 什么是部分分式？

Partial fractions are the reverse of adding rational expressions. Given a single algebraic fraction with a factorisable denominator, we split it into a sum of simpler fractions called partial fractions. This is especially useful in integration, binomial expansion, and solving differential equations.

部分分式是代数分式相加的逆运算。给定一个分母可因式分解的单一代数分式，我们将其拆分为若干更简单的分式之和，这些简单分式称为部分分式。这在积分、二项展开和微分方程求解中特别有用。

For example, 1/[(x+1)(x+2)] can be written as 1/(x+1) − 1/(x+2), which is much easier to integrate term by term.

例如，1/[(x+1)(x+2)] 可以写成 1/(x+1) − 1/(x+2)，这样逐项积分要容易得多。

2. The Key Idea: Denominator Factors | 核心思想：分母因式

Before splitting, factorise the denominator completely into linear and, where necessary, irreducible quadratic factors. The form of the partial fractions depends on the types of factors present: distinct linear factors, repeated linear factors, and irreducible quadratic factors.

拆分之前，先将分母完全因式分解为线性因式，必要时还包括不可约二次因式。部分分式的形式取决于因式类型：不同线性因式、重复线性因式以及不可约二次因式。

Always check the degree of the numerator. If the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is improper and must be divided first.

始终检查分子的次数。如果分子的次数大于或等于分母的次数，则该分式为假分式，必须先进行多项式除法。

3. Distinct Linear Factors | 不同线性因式

Suppose the denominator factorises into distinct linear factors, for example (x+1)(x−2). We write:

假设分母可因式分解为不同线性因式，例如 (x+1)(x−2)。我们设：

f(x)/[(x+1)(x−2)] ≡ A/(x+1) + B/(x−2

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