📚 2B Partial Fractions | 2B 部分分式分解
Partial fractions is an essential algebraic technique in IB Mathematics, particularly useful for integrating rational functions, solving differential equations, and simplifying complex expressions. The core idea is to express a complicated rational function as a sum of simpler fractions, each having a denominator that is a factor of the original denominator. In the IB Analysis & Approaches HL (and some applications in AI HL) curriculum, mastering partial fraction decomposition unlocks a wide range of calculus problems and deepens your understanding of polynomial identities.
部分分式分解是 IB 数学中一项核心代数技巧，尤其在积分有理函数、求解微分方程以及化简复杂表达式时非常有用。它的中心思想是将一个复杂的有理函数表示成若干个简单分式的和，每个简单分式的分母都是原分母的一个因式。在 IB 分析与解释高阶课程（以及部分 AI 高阶应用）中，掌握部分分式分解能够帮助你解决大量微积分问题，并加深对多项式恒等式的理解。
1. Introduction to Partial Fractions | 部分分式简介
A rational function is a ratio of two polynomials, written as P(x)/Q(x). Partial fraction decomposition applies when the degree of the numerator P(x) is less than the degree of the denominator Q(x) – this is called a proper rational function. The goal is to rewrite P(x)/Q(x) as a sum of fractions with simpler denominators, such as A/(x-a) or (Bx+C)/(ax²+bx+c). This decomposition makes it much easier to integrate or manipulate the original expression.
有理函数是两个多项式之比，记作 P(x)/Q(x)。当分子多项式 P(x) 的次数小于分母多项式 Q(x) 的次数时，称为真分式，此时就可以进行部分分式分解。我们的目标是将 P(x)/Q(x) 重新写成若干个具有简单分母的分式之和，例如 A/(x-a) 或 (Bx+C)/(ax²+bx+c)。这种分解能够大大简化原表达式的积分与运算。
2. Proper and Improper Fractions | 真分式与假分式
Before decomposing, check if the rational function is proper. If the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is improper. In such cases, you must first perform polynomial long division to express the function as a polynomial plus a proper fraction. For instance, (x³)/(x²-1) becomes x + x/(x²-1) after division, and only the proper part x/(x²-1) is decomposed further. Skipping this step is a common error that leads to incorrect decompositions.
在分解之前，务必检查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，则该分式为假分式。此时必须先进行多项式长除法，将函数表示成一个多项式加上一个真分式。例如，(x³)/(x²-1) 经长除后可写成 x + x/(x²-1)，而后仅对其中的真分式 x/(x²-1) 进行分解。忽略这一步是常见错误，往往会导致分解完全错误。
3. Factorization of the Denominator | 分母的因式分解
The first systematic step in partial fractions is to factor the denominator Q(x) completely into linear and irreducible quadratic factors over the real numbers. Linear factors are expressions like (x - r), while irreducible quadratic factors have the form ax²+bx+c with a negative discriminant (b²-4ac < 0). The form of the partial fraction expansion depends entirely on the nature of these factors. Working with factored denominators also makes it easier to solve for the unknown constants.
部分分式分解的第一步是将分母 Q(x) 在实数范围内完全分解为线性因式和不可约二次因式。线性因式形如 (x - r)，而不可约二次因式形如 ax²+bx+c，其判别式 b²-4ac < 0。部分分式的具体展开形式完全取决于这些因式的类型。将分母写成因式相乘的形式也有利于后续求解未知常数。
4. Distinct Linear Factors | 不同线性因式
n decomposition has the form A₁/(x-a₁) + A₂/(x-a₂) + ... + Aₙ/(x-aₙ). Each constant Aᵢ can be found by clearing denominators and solving a system of equations, or by using the convenient "cover-up" e remaining expression. This works quickly when all linear factors are distinct and non-repeating.
当分母分解为互不相同的一次因式，例如 (x-a₁)(x-a₂)...(x-aₙ) 时，部分分式分解的形式为 A₁/(x-a₁) + A₂/(x-a₂) + ... + Aₙ/(x-aₙ)。每一个常数 Aᵢ 都可以通过去分母后解方程组求得，也可以使用便捷的“遮盖法”。遮盖法的操作是：要求 Aᵢ 时，遮住分母中的因式 (x-aᵢ)，然后将 x = aᵢ 代入剩下的表达式中即可直接得到 Aᵢ。当所有线性因式都不相同时，这种方法非常快捷。
5. Worked Example: Distinct Linear Factors | 示例：不同线性因式
d 2A - B = 3. Solving, we find A = 8/3 and B = 7/3. Thus, the decomposition is (8/3)/(x-1) + (7/3)/(x+2).
试分解 (5x+3)/((x-1)(x+2))。首先写出假设形式：(5x+3)/((x-1)(x+2)) = A/(x-1) + B/(x+2)。两边同乘公分母 (x-1)(x+2) 得到 5x+3 = A(x+2) + B(x-1)。展开并合并同类项可得 5x+3 = (A+B)x + (2A - B)。比较系数得到方程组 A + B = 5，2A - B = 3。解得 A = 8/3，B = 7/3。因此分解结果为 (8/3)/(x-1) + (7/3)/(x+2)。
6. Repeated Linear Factors | 重复线性因式
If the denominator contains a repeated linear factor (x-a)ⁿ, the partial fraction expansion must include a series of fractions for that factor: A₁/(x-a) + A₂/(x-a)² + ... + Aₙ/(x-a)ⁿ. The degree n corresponds to the multiplicity of the root. For example, a factor (x-1)³ yields three terms in the decomposition. All constants for the repeated factor cannot be found by the simple cover-up method alone; you generally need to compare coefficients or substitute convenient values of x after clearing denominators.
如果分母含有重复的线性因式 (x-a)ⁿ，那么部分分式展开中必须为该因式设置一系列分式：A₁/(x-a) + A₂/(x-a)² + ... + Aₙ/(x-a)ⁿ。这里的指数 n 对应该重根的重复次数。例如，因式 (x-1)³ 对应展开式中的三项。重复因式对应的全部常数不能仅靠简单的遮盖法求得，通常需要在去分母后通过比较系数或代入方便的 x 值来求解。
7. Worked Example: Repeated Linear Factor | 示例：重复线性因式
nt: -2A + 2B + C = 5. Solving gives A = -1/9, B = 8/3, C = 1/9. The decomposition is -1/(9(x-1)) + 8/(3(x-1)²) + 1/(9(x+2)).
试分解 (3x+5)/((x-1)²(x+2))。分解形式必须设为：A/(x-1) + B/(x-1)² + C/(x+2)。两边乘以整个分母得 3x+5 = A(x-1)(x+2) + B(x+2) + C(x-1)²。细致展开：(x-1)(x+2) = x² + x - 2，所以 A(x²+x-2) + B(x+2) + C(x² - 2x +1) = 3x+5。合并同类项：x² 项系数 A + C = 0；x 项系数 A + B - 2C = 3；常数项 -2A + 2B + C = 5。解该方程组得 A = -1/9, B = 8/3, C = 1/9。最终分解为 -1/(9(x-1)) + 8/(3(x-1)²) + 1/(9(x+2))。
8. Irreducible Quadratic Factors | 不可约二次因式
When the denominator has an irreducible quadratic factor ax²+bx+c (where b²-4ac < 0), the corresponding term in the partial fraction expansion is of the form (Bx+C)/(ax²+bx+c). The numerator is a linear polynomial in x, not just a constant. This is necessary because the quadratic cannot be split into real linear factors. If the quadratic factor is repeated, say (ax²+bx+c)ⁿ, then the expansion includes terms (B₁x+C₁)/(ax²+bx+c) + (B₂x+C₂)/(ax²+bx+c)² + ... up to multiplicity n.
当分母含有不可约二次因式 ax²+bx+c（满足 b²-4ac < 0）时，对应的部分分式展开项应形如 (Bx+C)/(ax²+bx+c)。注意分子是 x 的一次多项式，而不仅仅是一个常数。这是因为此二次式无法在实数范围内进一步分解为一次因式。若二次因式出现重复，例如 (ax²+bx+c)ⁿ，则展开式中需包含 (B₁x+C₁)/(ax²+bx+c) + (B₂x+C₂)/(ax²+bx+c)² + ...，直至重复次数 n。
9. Worked Example: Irreducible Quadratic Factor | 示例：不可约二次因式
B = 2, C = 0. Hence, the decomposition is -2/(x+1) + 2x/(x²+1). Notice the numerator constant C turned out to be zero in this case, but it must always be included in the setup.
考虑 (2x-3)/((x+1)(x²+1))。x²+1 在实数范围内不可约。分解形式设为 A/(x+1) + (Bx+C)/(x²+1)。去分母得 2x-3 = A(x²+1) + (Bx+C)(x+1)。展开：2x-3 = A x² + A + B x² + Bx + Cx + C。合并同类项：x² 系数：A + B = 0；x 系数：B + C = 2；常数项：A + C = -3。解得 A = -2, B = 2, C = 0。因此分解结果为 -2/(x+1) + 2x/(x²+1)。虽然本例中分子常数 C 最终为零，但设立表达式时仍必须将其包含进去。
10. Using Partial Fractions in Integration | 部分分式在积分中的应用
) = 5/2/(x-3) + 3/2/(x+1). Then the integral becomes (5/2)∫ 1/(x-3) dx + (3/2)∫ 1/(x+1) dx, which evaluates to (5/2)ln|x-3| + (3/2)ln|x+1| rators appear, integration may involve completing the square and using arctan or ln forms.
IB 数学中学习部分分式的主要目的之一就是能够对那些无法通过简单换元求解的有理函数进行积分。例如，计算 ∫ (4x-2)/(x²-2x-3) dx，先将分母因式分解为 (x-3)(x+1) 并对被积函数进行部分分式分解：(4x-2)/((x-3)(x+1)) = 5/2/(x-3) + 3/2/(x+1)。于是积分化为 (5/2)∫ 1/(x-3) dx + (3/2)∫ 1/(x+1) dx，结果为 (5/2)ln|x-3| + (3/2)ln|x+1| + C。当出现带二次分子的项时，积分可能需要通过配方并利用 arctan 或 ln 等形式完成。
11. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
Students often forget to perform polynomial long division on improper fractions, leading to a false decomposition. Another frequent error is using an incorrect form for repeated or quadratic factors – for a squared linear factor (x-1)², missing the term with denominator (x-1) (not just (x-1)²) will make the system inconsistent. Also, algebraic mistakes when expanding and collecting like terms can give wrong constants. Always check your decomposition by recombining the partial fractions to see if you recover the original rational function. In exam settings, a quick verification can save valuable marks.
学生们常常忘记对假分式先做多项式长除法，直接分解导致完全错误。另一个常见错误是在处理重复因式或二次因式时设错形式——例如对于 (x-1)²，漏掉了分母为 (x-1) 的项（而不仅仅设 (x-1)² 的项）会使方程组无解或解得错误常数。此外，展开与合并同类项时的代数错误也会导致常数算错。分解完成后，务必通过重新合并你的部分分式来验证是否能够还原到原来的有理函数。在考试中，快速验算可以保住宝贵的分数。
12. Summary and Key Takeaways | 总结与要点
Partial fractions break a complex rational expression into simpler building blocks. The procedure can be summarised as: (1) ensure the fraction is proper, performing division if needed; (2) factor the denominator completely; (3) write the correct form based on distinct linear, repeated linear, and irreducible quadratic factors; (4) clear denominators and solve for the unknown constants; (5) use the result for integration or other operations. Mastery of this technique gives you a powerful tool for IB calculus, and with careful practice, the process becomes systematic and reliable.
部分分式分解把一个复杂的有理式拆解为简单的分式组合。整个过程可以总结为：(1) 确保分式为真分式，必要时进行长除法；(2) 将分母完全分解因式；(3) 根据不同线性因式、重复线性因式以及不可约二次因式写出正确展开形式；(4) 去分母并求出未知常数；(5) 将分解结果应用于积分或其他运算。掌握了这一技巧，你就拥有了解决 IB 微积分问题的强大工具；通过细致练习，整个过程会变得系统且可靠，成为你的得分利器。
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