📚 Partial Fractions | 部分分式
In A-Level Mathematics, partial fractions are a vital algebraic technique used to simplify complex rational expressions into a sum of simpler fractions. This method is particularly useful when integrating rational functions, solving differential equations, or working with Laplace transforms in further studies. Mastering partial fractions allows you to break down expressions with polynomial denominators into components that are much easier to manipulate. In the Edexcel syllabus, the focus is on proper fractions where the denominator can be factorised into linear factors, including repeated linear factors, and occasionally simple quadratic factors that cannot be reduced further over the real numbers. This article will guide you through the essential concepts, step-by-step methods, and provide plenty of worked examples to build your confidence and accuracy.
在 A-Level 数学中，部分分式是一项重要的代数技巧，用于将复杂的有理式分解为几个简单分式的和。这种方法在积分有理函数、解微分方程，甚至高等课程中的拉普拉斯变换等场景都非常有用。掌握部分分式能让你把分母为多项式的表达式拆分成更容易处理的部分。Edexcel 大纲重点考察真分式，其分母可分解为线性因式（包括重复线性因式），偶尔也会出现无法在实数范围内进一步分解的简单二次因式。本文将带你梳理核心概念，逐步讲解方法，并提供大量范例，帮你建立信心，提升解题准确性。
1. What Are Partial Fractions? | 什么是部分分式？
A partial fraction decomposition rewrites a single rational expression as a sum of two or more simpler rational expressions. For example, the fraction 3/(x² − 1) can be expressed as A/(x − 1) + B/(x + 1), where A and B are constants to be determined. This technique is only applicable when the degree of the numerator is strictly less than the degree of the denominator – a condition known as a proper fraction. If you encounter an improper fraction, you must first perform polynomial long division to obtain a polynomial plus a proper fraction, and then decompose the proper part.
部分分式分解就是把一个有理式重新写成两个或多个更简单有理式的和。例如，分式 3/(x² − 1) 可以表示为 A/(x − 1) + B/(x + 1)，其中 A 和 B 是待确定的常数。只有当分子的次数严格低于分母的次数时，也就是真分式，才能直接使用该技巧。如果碰见假分式，你必须先用多项式长除法求出一个多项式与一个真分式之和，再对真分式部分进行分解。
2. Denominator Factorisation | 分母因式分解
Before setting up a partial fraction form, the denominator must be fully factorised into irreducible polynomials over the real numbers. For linear factors, you look for expressions of the form (ax + b). A quadratic factor like x² + 1 is irreducible over the reals and is treated as a single factor. The Edexcel exam mainly tests your skill with distinct and repeated linear factors. Always ensure the denominator is fully factorised, cancelling any common factors with the numerator first if possible, to simplify the expression before decomposition.
在设定部分分式的形式之前，分母必须被完全分解为实数范围内的不可约多项式。对于线性因式，你要寻找形如 (ax + b) 的因子。像 x² + 1 这样的二次式在实数范围不可约，会被当做一个整体因式来处理。Edexcel 考试主要考查你对不同和重复线性因式的处理能力。一定要确保分母已被彻底分解，可能的话先约去分子与分母的公因式，使表达式化简后再做分解。
3. Linear Factors (Distinct) | 不同线性因式
When the denominator consists of distinct linear factors, the general form for each linear factor (ax + b) is a constant numerator A divided by that factor. For instance, given a proper fraction with denominator (2x + 1)(x − 3), we set up the identity: P(x)/[(2x + 1)(x − 3)] ≡ A/(2x + 1) + B/(x − 3). The constants A and B are then found by either substituting convenient values of x that eliminate one term, or by equating coefficients of like powers of x after multiplying through by the denominator. The substitution method is often faster for distinct linear factors.
当分母由不同的线性因式组成时，每个线性因式 (ax + b) 对应的部分分式形式为常数 A 除以该因式。例如，给定一个真分式，分母为 (2x + 1)(x − 3)，我们可以建立恒等式：P(x)/[(2x + 1)(x − 3)] ≡ A/(2x + 1) + B/(x − 3)。然后通过代入特殊的 x 值消去某一项，或者将等式两边乘以分母后比较同次幂的系数，来求出常数 A 和 B。对于不同线性因式，代入法通常更快。
4. Worked Example – Two Distinct Linear Factors | 范例 – 两个不同线性因式
Express (5x + 1)/[(x − 1)(x + 2)] in partial fractions.
将 (5x + 1)/[(x − 1)(x + 2)] 写成部分分式。
Step 1: Write the identity: (5x + 1)/[(x − 1)(x + 2)] ≡ A/(x − 1) + B/(x + 2).
步骤1：写出恒等式：(5x + 1)/[(x − 1)(x + 2)] ≡ A/(x − 1) + B/(x + 2)。
Step 2: Multiply both sides by (x − 1)(x + 2): 5x + 1 ≡ A(x + 2) + B(x − 1).
步骤2：两边同时乘以 (x − 1)(x + 2)：5x + 1 ≡ A(x + 2) + B(x − 1)。
Step 3: Choose x = 1 to eliminate B: 5(1) + 1 = A(1 + 2) ⇒ 6 = 3A ⇒ A = 2.
步骤3：取 x = 1 以消去 B：5(1) + 1 = A(1 + 2) ⇒ 6 = 3A ⇒ A = 2。
Step 4: Choose x = −2 to eliminate A: 5(−2) + 1 = B(−2 − 1) ⇒ −9 = −3B ⇒ B = 3.
步骤4：取 x = −2 以消去 A：5(−2) + 1 = B(−2 − 1) ⇒ −9 = −3B ⇒ B = 3。
Step 5: Write the final decomposition: (5x + 1)/[(x − 1)(x + 2)] ≡ 2/(x − 1) + 3/(x + 2).
步骤5：写出最终分解式：(5x + 1)/[(x − 1)(x + 2)] ≡ 2/(x − 1) + 3/(x + 2)。
∫ [2/(x − 1) + 3/(x + 2)] dx = 2 ln|x − 1| + 3 ln|x + 2| + C
5. Repeated Linear Factors | 重复线性因式
When a linear factor repeats n times, say (ax + b)ⁿ, the partial fraction decomposition must include a sequence of fractions with constant numerators and denominators of increasing powers of that factor. The general pattern is: A₁/(ax + b) + A₂/(ax + b)² + ... + Aₙ/(ax + b)ⁿ. This ensures you capture all possible terms when recombining to the original rational expression. For example, if the denominator contains (x − 3)², the setup must include both a term with denominator (x − 3) and another with denominator (x − 3)². Neglecting the lower powers is a very common mistake.
当某个线性因式重复 n 次，比如 (ax + b)ⁿ，部分分式的分解式中必须包含一系列分母幂次递增的常数分子分式。一般形式为：A₁/(ax + b) + A₂/(ax + b)² + ... + Aₙ/(ax + b)ⁿ。这样在重新通分合并时，才能覆盖原表达式的所有可能项。例如，若分母含有 (x − 3)²，设定形式时必须既有一个分母为 (x − 3) 的项，也有一个分母为 (x − 3)² 的项。忽略低次幂项是非常常见的错误。
6. Worked Example – Repeated Linear Factor | 范例 – 重复线性因式
Decompose (4x + 1)/[(x − 2)²] into partial fractions.
将 (4x + 1)/[(x − 2)²] 分解为部分分式。
Setup: (4x + 1)/(x − 2)² ≡ A/(x − 2) + B/(x − 2)².
设定形式：(4x + 1)/(x − 2)² ≡ A/(x − 2) + B/(x − 2)²。
Multiply through by (x − 2)²: 4x + 1 ≡ A(x − 2) + B.
两边乘以 (x − 2)²：4x + 1 ≡ A(x − 2) + B。
Expand: 4x + 1 ≡ Ax − 2A + B.
展开：4x + 1 ≡ Ax − 2A + B。
Equate coefficients: For x: A = 4. Constant term: −2A + B = 1 ⇒ −8 + B = 1 ⇒ B = 9.
比较系数：x 项：A = 4。常数项：−2A + B = 1 ⇒ −8 + B = 1 ⇒ B = 9。
Thus, (4x + 1)/(x − 2)² ≡ 4/(x − 2) + 9/(x − 2)².
因此，(4x + 1)/(x − 2)² ≡ 4/(x − 2) + 9/(x − 2)²。
∫ [4/(x − 2) + 9/(x − 2)²] dx = 4 ln|x − 2| − 9/(x − 2) + C
7. Handling Quadratic Factors | 二次因式的处理
In the Edexcel spec, you may occasionally need to deal with an irreducible quadratic factor like (x² + 1). For such a factor, the partial fraction form is (Ax + B)/(x² + 1), where the numerator is linear rather than constant. The constants are found by comparing coefficients, often combined with substitution if other linear factors exist. However, this appears less frequently, and you can focus your revision on linear denominators, as these form the core of the exam questions.
在 Edexcel 大纲中，你偶尔可能需要处理像 (x² + 1) 这样的不可约二次因式。对于这类因式，部分分式的形式为 (Ax + B)/(x² + 1)，分子为一次式而非常数。常数可通过比较系数求得，如果还含有其他线性因式，常与代入法结合求解。但这部分考得较少，可以将复习重点放在线性因式的分母上，这是考试题的核心。
8. Algebraic Long Division First | 先进行代数长除法
Before applying partial fractions, always check if the rational expression is improper. An improper fraction has a numerator with degree greater than or equal to the denominator. For example, (x³ + 2x)/(x² − 1) is improper. Perform polynomial long division to write it as Q(x) + R(x)/D(x), where R(x)/D(x) is proper. Then decompose R(x)/D(x) into partial fractions. Missing this step will lead to an invalid decomposition and loss of marks.
在应用部分分式之前，一定要检查有理式是否为假分式。假分式的分子次数大于或等于分母次数。例如，(x³ + 2x)/(x² − 1) 就是假分式。先进行多项式长除法，将其写成 Q(x) + R(x)/D(x) 的形式，其中 R(x)/D(x) 是真分式。然后再把 R(x)/D(x) 分解为部分分式。遗漏这一步将导致无效的分解并丢分。
tract: (x³ + 2x) − (x³ − x) = 3x. So result = x + 3x/(x² − 1). Then decompose 3x/(x² − 1).
示例：(x³ + 2x)/(x² − 1)。除法：x³ ÷ x² = x。乘：x(x² − 1) = x³ − x。相减：(x³ + 2x) − (x³ − x) = 3x。所以结果为 x + 3x/(x² − 1)。再对 3x/(x² − 1) 进行部分分式分解。
9. The Method of Equating Coefficients | 比较系数法
or when repeated factors demand more equations. After clearing denominators, expand the right-hand side and collect like terms. Then match the coefficients of xⁿ on both sides to form a system of simultaneous equations. Solve for A, B, C, etc. For example, with A(x + 3) + B(x − 2) ≡ 7x − 1, expanding gives (A + B)x + (3A − 2B) ≡ 5. The method is systematic and works for all cases, though it can be algebraically heavy.
当难以找到方便的代入值，或者重复因式需要更多方程时，比较系数法是可靠的选择。去掉分母后，展开右边并合并同类项，然后让等式两边 xⁿ 的系数对应相等，形成联立方程组。解出 A、B、C 等常数。例如，由 A(x + 3) + B(x − 2) ≡ 7x − 1，展开得 (A + B)x + (3A − 2B) ≡ 7x − 1，于是得到 A + B = 7 和 3A − 2B = −1。解得 A = 13/5，B = 22/5。该方法系统且适用于所有情况，但代数运算量可能较大。
10. Common Mistakes to Watch For | 常见错误提醒
Top errors include forgetting to set up all terms for repeated factors, incorrectly assuming a numerator is constant for a quadratic factor, failing to perform division for improper fractions, and making arithmetic slip-ups when solving for constants. Always double-check that your final decomposition recombines to the original expression by finding a common denominator. Another pitfall is missing a factor: ensure the denominator is fully factorised, taking out any common factor like 2 from (2x + 4) to 2(x + 2) first if it simplifies the workflow.
最常见的错误包括：忘记为重复因式设定所有必要的项、误以为二次因式对应的是常数分子、对假分式未做除法，以及求解常数时出现算术失误。务必通过通分检验最终分解式是否还原为原表达式。另一个陷阱是遗漏因子：确保分母已被彻底分解，比如先把 (2x + 4) 提取公因数 2 变成 2(x + 2)，如果这样能简化后续步骤的话。
11. Application in Integration | 在积分中的应用
s decomposed into simpler fractions, standard integrals such as ∫ 1/(ax + b) dx = (1/a) ln|ax + b| ure mathematics papers and often carries a high mark weight when combined with a subsequent integration task or area calculation.
部分分式主要用于积分有理函数。表达式一旦被分解为更简单的分式，就可以使用标准积分公式，如 ∫ 1/(ax + b) dx = (1/a) ln|ax + b| 以及 ∫ 1/(ax + b)ⁿ dx 等。这样就把看似复杂的被积函数转化为容易处理的分项。在 Edexcel 纯数试卷中，这是一个标准题型，当后续再结合积分运算或面积计算时，通常占分较高。
12. Summary and Revision Tips | 总结与复习建议
Mastering partial fractions demands careful algebraic manipulation and a clear structured approach. Memorise the standard forms for distinct linear factors, repeated linear factors, and the linear-over-quadratic case. Practice identifying when to use substitution versus coefficient equating. Always start by checking if the fraction is proper. A solid workout of past paper questions will sharpen your ability to spot the fastest method. Keep your work neat, and verify by adding the partial fractions back together mentally or on scratch paper to catch errors early.
掌握部分分式需要细致的代数操作和清晰的结构化思路。熟记不同线性因式、重复线性因式以及一次分子除以二次因式的标准形式。练习判断何时用代入法、何时用比较系数法。始终以检查分式是否为真分式作为第一步。大量练习往年真题能帮助你快速捕捉最优解法。保持书写整洁，并通过心算或在草稿纸上将部分分式重新相加来验算，及早发现错误。
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