📚 Partial Fractions Decomposition | 部分分式分解
In Edexcel A-Level Pure Mathematics, partial fractions offer a systematic method for rewriting complex rational expressions as a sum of simpler fractions. This technique is essential for integration, solving differential equations, and simplifying algebraic manipulations. Mastering decomposition ensures you can break down awkward quotients into components that are easy to integrate or analyse.
在Edexcel A-Level纯数学中，部分分式提供了一种系统的方法，将复杂的有理表达式改写为几个简单分式的和。该技巧对积分、求解微分方程以及简化代数运算至关重要。掌握分解方法能确保你将晦涩的分式拆解为易于积分或分析的组成部分。
1. What Are Partial Fractions? | 什么是部分分式
A rational function f(x)/g(x) can often be expressed as the sum of two or more simpler fractions whose denominators are factors of g(x). This reverse process of combining fractions is called partial fractions decomposition. For example, the single fraction (2x+1)/((x+1)(x-2)) can be split into A/(x+1) + B/(x-2) for suitable constants A and B.
有理函数 f(x)/g(x) 通常可表示为两个或更多较简单分式的和，这些分式的分母正好是 g(x) 的因式。这种将分式“拆开”的逆过程称为部分分式分解。例如，单一分式 (2x+1)/((x+1)(x-2)) 可以拆分为 A/(x+1) + B/(x-2)，其中 A 和 B 为适当常数。
2. When to Use Partial Fractions | 何时使用部分分式
Partial fractions are used when you need to integrate a rational function where the denominator factorises, or when simplifying expressions in series expansions and inverse Laplace transforms. In Edexcel exams, the most common context is integration: splitting a fraction into simpler parts that integrate to natural logarithms, arctangents, or powers.
当需要对分母可因式分解的有理函数进行积分，或者在级数展开和拉普拉斯逆变换中简化表达式时，就会用到部分分式。Edexcel考试中最常见的场景是积分：把分式拆成可直接积出自然对数、反正切或幂函数的简单部分。
3. Proper vs Improper Fractions | 真分式与假分式
A rational expression is proper if the degree of the numerator is less than the degree of the denominator. If it is improper, polynomial long division must be performed first to obtain a polynomial part plus a proper remainder. The partial fractions technique is then applied to the proper remainder.
如果分子的次数低于分母的次数，该有理式就是真分式。如果是假分式，必须先进行多项式长除法，得到一个多项式部分加上一个真余式。随后才对真余式应用部分分式方法。
Example: (x³+2x²)/(x²-1) → x+2 + (x+2)/(x²-1)
示例：(x³+2x²)/(x²-1) → x+2 + (x+2)/(x²-1)
4. Distinct Linear Factors | 不同线性因式
When the denominator factorises into distinct linear factors (x-a₁)(x-a₂)...(x-aₙ), the decomposition takes the form A₁/(x-a₁) + A₂/(x-a₂) + ... + Aₙ/(x-aₙ). Each constant is found by clearing denominators and either substituting convenient x-values or equating coefficients.
当分母分解为互不相同的线性因式 (x-a₁)(x-a₂)...(x-aₙ) 时，分解形式为 A₁/(x-a₁) + A₂/(x-a₂) + ... + Aₙ/(x-aₙ)。通过去分母并代入方便的 x 值或比较系数即可求出各个常数。
(3x+5)/((x-1)(x+2)) = A/(x-1) + B/(x+2)
(3x+5)/((x-1)(x+2)) = A/(x-1) + B/(x+2)
5. Repeated Linear Factors | 重复线性因式
If a linear factor (x-a) appears r times, the decomposition must include terms A₁/(x-a) + A₂/(x-a)² + ... + Aᵣ/(x-a)ʳ. This ensures every possible power up to the multiplicity is covered. Edexcel questions often feature a repeated linear factor alongside a distinct linear factor.
如果线性因式 (x-a) 出现 r 次，分解中必须包含 A₁/(x-a) + A₂/(x-a)² + ... + Aᵣ/(x-a)ʳ 各项。这确保覆盖了直到该重数的所有可能幂次。Edexcel题目常会同时出现一个重复线性因式和一个不同线性因式。
(4x+1)/((x-3)²(x+1)) = A/(x-3) + B/(x-3)² + C/(x+1)
(4x+1)/((x-3)²(x+1)) = A/(x-3) + B/(x-3)² + C/(x+1)
6. Irreducible Quadratic Factors | 不可约二次因式
When the denominator contains a quadratic factor that cannot be factorised over the reals, such as x²+bx+c with discriminant Δ<0, the corresponding numerator must be of the linear form Ax+B. This ensures the decomposition handles the quadratic correctly and sets up integration leading to arctan or log forms.
当分母含有无法在实数范围内分解的二次因式（例如判别式 Δ<0 的 x²+bx+c），对应的分子必须是线性形式 Ax+B。这样分解才能正确应对二次式，并为积分时产生反正切或对数形式做好准备。
(2x²+3)/((x+1)(x²+1)) = A/(x+1) + (Bx+C)/(x²+1)
(2x²+3)/((x+1)(x²+1)) = A/(x+1) + (Bx+C)/(x²+1)
7. Substitution Method for Constants | 代入法求常数
s individual constants. For example, substituting x=1, x=-2 eliminates all but one term. This method is quick and minimises algebra, but it cannot directly find constants for repeated or quadratic factors.
写出一般分解式后，两侧同乘分母。选择能使特定因式为零的 x 值，可隔离出单个常数。例如代入 x=1、x=-2 可以消去除了某一项之外的所有项。这种方法快捷且代数运算量小，但无法直接求出重复因式或二次因式对应的常数。
Let x=1 to find A, then x=-2 to find B in (3x+5)/((x-1)(x+2)).
在 (3x+5)/((x-1)(x+2)) 中令 x=1 求 A，再令 x=-2 求 B。
8. Equating Coefficients Method | 比较系数法
Expand the right-hand side after clearing denominators and collect coefficients of like powers of x. Equate these to the coefficients in the original numerator to form simultaneous equations. This systematic approach works for all factor types, especially when substitution cannot provide all constants directly.
去分母后展开右侧，并按 x 的幂次合并系数。将这些系数与原分子中的系数对应相等，形成联立方程组。这种系统化的方法适用于所有因式类型，尤其是在代入法无法直接给出全部常数时。
For (Bx+C)/(x²+1) terms, compare coefficients of x² and constants.
对于 (Bx+C)/(x²+1) 项，需比较 x² 和常数项的系数。
9. Handling Improper Fractions | 处理假分式
If the numerator degree is greater than or equal to the denominator degree, perform long division first. The quotient becomes a polynomial, and the remainder becomes the new proper fraction. Apply partial fractions to the remainder only. Many Edexcel past paper questions test this initial step before decomposition.
若分子次数大于或等于分母次数，需先进行长除法。商式成为多项式部分，余数成为新的真分式。仅对余数部分使用部分分式。Edexcel 历年真题中许多题目都会考察这个分解前的初始步骤。
(x³+2)/(x²-1) = x + (x+2)/(x²-1) → decompose (x+2)/(x²-1).
(x³+2)/(x²-1) = x + (x+2)/(x²-1) → 对 (x+2)/(x²-1) 进行分解。
10. Integration Using Partial Fractions | 利用部分分式积分
pplication is integration. After decomposition, terms of the form ∫ A/(x-a) dx become A ln|x-a| ield arctan forms after completing the square. Always include the constant of integration.
Edexcel 的主要应用是积分。分解后，形如 ∫ A/(x-a) dx 的项积分为 A ln|x-a|。带有二次线性因式的项积分得到 -A/(x-a)。二次分母常常通过配方法导出反正切形式。务必加上积分常数。
∫ (2x+1)/((x+1)(x-2)) dx = ∫ [1/(x+1) + 1/(x-2)] dx = ln|x+1| + ln|x-2| + c
∫ (2x+1)/((x+1)(x-2)) dx = ∫ [1/(x+1) + 1/(x-2)] dx = ln|x+1| + ln|x-2| + c
11. Summary of Factor Types and Form | 因式类型与分解形式总结
The table below summarises the decomposition format required for each denominator factor type. Recognising the form instantly is key to efficient exam performance.
下表概括了每种分母因式类型所需的分解格式。能迅速识别形式是提高考试效率的关键。
| Factor Type | Form of Decomposition |
| Distinct linear (x-a) | A/(x-a) |
| Repeated linear (x-a)ⁿ | A₁/(x-a) + A₂/(x-a)² + ... + Aₙ/(x-a)ⁿ |
| Irreducible quadratic ax²+bx+c | (Ax+B)/(ax²+bx+c) |
12. Common Mistakes and Essential Tips | 常见错误与重要提示
Forgetting long division when the fraction is improper is a frequent mistake. Also, always check that the numerator for a quadratic factor is written as Ax+B, not just a constant. When integrating, remember the absolute value inside the logarithm. Practise with a mixture of factor types to build speed and accuracy.
最常见的错误是当分式为假分式时忘记先进行长除法。此外，务必检查二次因式所对应的分子是否写为 Ax+B，而非只写常数。积分时，记得对数里要加绝对值。混合练习各种因式类型，以培养速度和准确度。
Published by TutorHao | Pure Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图