📚 Partial Fractions | 部分分式
Partial fractions are a powerful algebraic tool that allow us to express a complicated rational expression as a sum of simpler fractions. In A-Level Edexcel Mathematics, mastering partial fractions is essential not only for algebraic manipulation but also for integration, solving differential equations, and working with series expansions. This technique reverses the process of adding algebraic fractions, breaking a single fraction with a factorable denominator into a sum of fractions with those factors as denominators.
部分分式是一种强大的代数工具，可以把复杂的 rational expression（有理式）转化为几个更简单分式的和。在 Edexcel A-Level 数学中，掌握部分分式不仅对代数操作至关重要，对积分、解微分方程以及级数展开也同样关键。这一方法相当于对分式加法进行逆向操作，将分母可分解因式的单个分式拆分为若干个以这些因式为分母的分式之和。
1. What Are Partial Fractions? | 什么是部分分式？
Given a rational function P(x)/Q(x), where P(x) and Q(x) are polynomials and the degree of P(x) is less than the degree of Q(x), we can decompose it into a sum of simpler fractions called partial fractions. Each partial fraction has a denominator that is a factor of Q(x) and a numerator of lower degree than that factor. This decomposition makes it much easier to integrate, differentiate, or expand the original expression.
对于一个有理函数 P(x)/Q(x)，其中 P(x) 和 Q(x) 都是多项式且 P(x) 的次数低于 Q(x) 的次数，我们可以将其分解成若干个更简单分式的和，即 partial fractions。每一个部分分式的分母都是 Q(x) 的一个因式，且分子的次数低于该因式的次数。这种分解能让原式的积分、微分或展开变得非常简单。
2. Proper and Improper Rational Functions | 真分式与假分式
. If it is improper (numerator degree ≥ denominator degree), we first perform polynomial long division to rewrite it as a polynomial plus a proper rational remainder. Only the proper part is then split into partial fractions.
分解之前，必须保证有理函数是真分式：分子的次数严格小于分母的次数。如果是假分式（分子的次数大于或等于分母的次数），我们首先使用多项式长除法，将其改写为一个多项式加上一个真分式余项。只有真分式部分才拆分成部分分式。
3. Linear Factors in the Denominator | 分母为一次因式
When the denominator factorises into distinct linear factors, such as (x - a)(x - b), the decomposition takes the form A/(x - a) + B/(x - b). The constants A and B are found by either equating coefficients or substituting convenient values of x. This is the simplest and most common type of partial fraction decomposition in A-Level exams.
当分母可分解为不同的一次因式时，例如 (x - a)(x - b)，分解形式为 A/(x - a) + B/(x - b)。常数 A 和 B 可以通过比较系数或者代入方便的 x 值来求得。这是 A-Level 考试中最简单也最常见的部分分式类型。
4. Worked Example: Distinct Linear Factors | 示例：不同的一次因式
Decompose (5x + 1) / ((x - 2)(x + 3)) into partial fractions.
将 (5x + 1) / ((x - 2)(x + 3)) 分解为部分分式。
A(5) → 11 = 5A → A = 11/5. Substitute x = -3: 5(-3) + 1 = B(-5) → -14 = -5B → B = 14/5. Hence the decomposition is (11/5)/(x - 2) + (14/5)/(x + 3).
我们设：(5x + 1) / ((x - 2)(x + 3)) ≡ A/(x - 2) + B/(x + 3)。两边同乘 (x - 2)(x + 3)：5x + 1 ≡ A(x + 3) + B(x - 2)。代入 x = 2：5(2) + 1 = A(5) → 11 = 5A → A = 11/5。代入 x = -3：5(-3) + 1 = B(-5) → -14 = -5B → B = 14/5。因此分解结果为 (11/5)/(x - 2) + (14/5)/(x + 3)。
5. Repeated Linear Factors | 重复的一次因式
If the denominator contains a repeated linear factor, say (x - a)², the decomposition must include terms for both (x - a)² and (x - a): A/(x - a) + B/(x - a)². More generally, for a factor (x - a)ⁿ, we include fractions with denominators (x - a), (x - a)², …, (x - a)ⁿ, each with a constant numerator.
如果分母中含有重复的一次因式，例如 (x - a)²，分解时必须同时包含 (x - a)² 和 (x - a) 的项：A/(x - a) + B/(x - a)²。更一般地，对于因式 (x - a)ⁿ，需要引入分母为 (x - a)、(x - a)²、…、(x - a)ⁿ 的分式，每个分式的分子都是常数。
6. Worked Example: Repeated Linear Factor | 示例：重复一次因式
Decompose (2x + 3) / ((x - 1)²) into partial fractions.
将 (2x + 3) / ((x - 1)²) 分解为部分分式。
ficients: for x: A = 2; constant: -A + B = 3 → -2 + B = 3 → B = 5. Thus the decomposition is 2/(x - 1) + 5/(x - 1)².
设：(2x + 3)/(x - 1)² ≡ A/(x - 1) + B/(x - 1)²。两边同乘 (x - 1)²：2x + 3 ≡ A(x - 1) + B。展开：2x + 3 ≡ Ax - A + B。比较系数：x 项：A = 2；常数项：-A + B = 3 → -2 + B = 3 → B = 5。因此分解为 2/(x - 1) + 5/(x - 1)²。
7. Irreducible Quadratic Factors | 不可约二次因式
When the denominator contains an irreducible quadratic factor such as ax² + bx + c (where the discriminant is negative), the corresponding partial fraction has a linear numerator: (Px + Q)/(ax² + bx + c). This factor cannot be further factorised over the real numbers, and the numerator must be a general linear expression.
如果分母含有不可约二次因式，例如 ax² + bx + c（其判别式小于零），对应的部分分式的分子应该是一个一次式：(Px + Q)/(ax² + bx + c)。这类因式在实数范围内不能再分解，分子必须是一般形式的一次表达式。
8. Worked Example: Quadratic Factor | 示例：二次因式
Decompose (3x² + 2x + 5) / ((x - 1)(x² + 1)) into partial fractions.
将 (3x² + 2x + 5) / ((x - 1)(x² + 1)) 分解为部分分式。
e: A x² + A + Bx² - Bx + Cx - C = (A + B)x² + (-B + C)x + (A - C). Compare coefficients: x²: A + B = 3; x: -B + C = 2; constant: A - C = 5. Solving yields A = 4, B = -1, C = 1. Thus, the decomposition is 4/(x - 1) + (-x + 1)/(x² + 1).
设：(3x² + 2x + 5)/[(x - 1)(x² + 1)] ≡ A/(x - 1) + (Bx + C)/(x² + 1)。两边相乘：3x² + 2x + 5 ≡ A(x² + 1) + (Bx + C)(x - 1)。右边展开：A x² + A + Bx² - Bx + Cx - C = (A + B)x² + (-B + C)x + (A - C)。比较系数：x²：A + B = 3；x：-B + C = 2；常数：A - C = 5。解得 A = 4，B = -1，C = 1。因此分解为 4/(x - 1) + (-x + 1)/(x² + 1)。
9. Using Partial Fractions in Integration | 部分分式在积分中的应用
One of the main reasons to study partial fractions in Edexcel A-Level is to integrate rational functions. After decomposition, each term becomes a standard integral: terms with linear denominators yield natural logarithms; terms with repeated linear factors produce inverse powers; and quadratic factors often lead to arctangent or combined logarithmic functions. This transforms a seemingly difficult integral into manageable pieces.
在 Edexcel A-Level 中学习部分分式的主要原因之一就是用来积分有理函数。分解之后，每一项都变成了标准积分：分母为一次式的项得到自然对数；重复一次因式项得到负幂次函数；二次因式往往导出反正切或组合对数函数。这样就把看似困难的积分化整为零，便于计算。
10. Common Mistakes to Avoid | 常见错误与避坑指南
Many students forget to check whether the fraction is improper before starting the decomposition, leading to an invalid partial fraction form. Another common error is setting up the wrong form for repeated or quadratic factors. Always match the numerator's degree to one less than the degree of the denominator factor. Additionally, double-check algebraic expansions and coefficient comparisons, as small sign errors can propagate through the entire solution.
很多学生在一开始忘记了检查分式是否为真分式，导致部分分式的形式无效。另一个常见错误是为重复因式或二次因式设置了错误的分解形式。务必保证分子的次数比分母因式的次数少一次。同时，要仔细核对代数展开和系数比较的过程，小小的符号错误都可能影响整个解答。
11. A-Level Exam Strategy | A-Level 考试策略
In the Edexcel A-Level exam, partial fractions frequently appear in integration questions, sometimes combined with the trapezium rule or differential equations. Always read the question carefully: it may ask you to express a rational function in partial fractions and then use the result to integrate or evaluate a sum. Show clear steps: state the form, multiply out, find constants, and rewrite the integral. Partial fractions can also help to evaluate sums of series by the method of differences.
在 Edexcel A-Level 考试中，部分分式经常出现在积分题里，有时与梯形法则或微分方程结合。一定要仔细审题：题目可能要求先把有理函数写成部分分式，再利用结果去积分或求值。解题步骤要清晰：写出分解形式、去分母、求常数、重写积分。部分分式还可以通过差消法帮助级数求和。
12. Summary Table of Decomposition Forms | 分解形式速查表
| Denominator Factor(s) | 分母因式 | Form of Partial Fractions | 部分分式形式 |
|---|---|
| Distinct linear: (x - a)(x - b) | A/(x - a) + B/(x - b) |
| Repeated linear: (x - a)² | A/(x - a) + B/(x - a)² |
| Repeated linear: (x - a)ⁿ | A₁/(x - a) + A₂/(x - a)² + … + Aₙ/(x - a)ⁿ |
| Irreducible quadratic: ax² + bx + c (Δ < 0) | (Px + Q)/(ax² + bx + c) |
| Mixed: (x - a)(x² + bx + c) | A/(x - a) + (Bx + C)/(x² + bx + c) |
This table summarises the standard setup for partial fractions based on denominator types. Memorising these forms will help you quickly identify the correct structure during the exam. Always check that you have included a term for every power of a repeated factor and that quadratic numerators are linear.
这张表格总结了基于分母类型的部分分式标准设定。记住这些形式可以帮助你在考试中快速识别正确的分解结构。务必检查是否已经为重复因式的每一次幂都设置了对应项，以及二次因式上的分子是否是一次式。
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