A-Level Mathematics: Partial Fraction Decomposition Methods | A-Level 数学：部分分式分解方法

📚 A-Level Mathematics: Partial Fraction Decomposition Methods | A-Level 数学：部分分式分解方法

Partial fraction decomposition is a fundamental algebraic technique in A-Level mathematics that expresses a rational function as a sum of simpler fractions. It serves as a bridge between algebraic manipulation and advanced topics such as integration and binomial expansion.

部分分式分解是 A-Level 数学中一项基础代数技术，它将一个有理函数表示为若干更简单分式之和。它是连接代数运算与积分、二项展开等进阶专题的重要桥梁。

1. Theoretical Foundation | 理论基础

The method applies to proper rational functions, where the degree of the numerator is less than the degree of the denominator. Every proper rational function can be uniquely decomposed into a sum of partial fractions whose denominators are derived from the factors of the original denominator.

该方法适用于真分式，即分子的次数低于分母的次数。每个真分式都可以唯一地分解为一系列部分分式之和，这些分式的分母由原分母的因式导出。

Consider a rational function of the form:

f(x) = P(x) / Q(x)

where Q(x) can be factored into linear factors and/or irreducible quadratic factors. The decomposition form depends entirely on the nature of these factors.

考虑形如下式的有理函数：

f(x) = P(x) / Q(x)

其中 Q(x) 可分解为线性因式和/或不可约二次因式。分解的具体形式完全取决于这些因式的性质。

2. Case 1: Distinct Linear Factors | 情况一：互异线性因式

When the denominator factors into distinct linear factors, each factor contributes one partial fraction term with a constant numerator. For example:

当分母分解为互不相同的线性因式时，每个因式对应一个分子为常数的部分分式项。例如：

5x – 3 / [(x – 1)(x + 2)] = A/(x – 1) + B/(x + 2)

To determine A and B, multiply both sides by the denominator (x – 1)(x + 2):

5x – 3 = A(x + 2) + B(x – 1)

Expanding and comparing coefficients of x and the constant term gives a system of linear equations. Alternatively, substitute x = 1 to eliminate the B term, giving A = 2/3; substitute x = -2 to give B = 13/3.

为确定 A 和 B，将等式两边同乘以分母 (x – 1)(x + 2)：

5x – 3 = A(x + 2) + B(x – 1)

展开并比较 x 的系数及常数项，可得线性方程组。另一种方法：代入 x = 1 消去 B 项，得 A = 2/3；代入 x = -2 可得 B = 13/3。

The substitution method is generally faster when all linear factors are distinct and real. Each zero of the denominator eliminates all but one unknown.

当所有线性因式互异且为实数时，代入法通常更为快捷。分母的每个零点都能消去除一个未知数外的所有项。

3. Case 2: Repeated Linear Factors | 情况二：重复线性因式

If a linear factor appears with multiplicity n, the decomposition must include n terms with increasing powers of that factor. For instance, for the denominator (x – 1)²(x + 3):

若某个线性因式以 n 重出现，则分解中必须包含该因式幂次递增的 n 个项。例如，对于分母 (x – 1)²(x + 3)：

(x² + 2x + 5) / [(x – 1)²(x + 3)] = A/(x – 1) + B/(x – 1)² + C/(x + 3)

Here, A/(x – 1) and B/(x – 1)² are both required; omitting the first term is a common error. After multiplying through by the full denominator, use a combination of substitution and coefficient comparison to solve for A, B, and C.

此处 A/(x – 1) 和 B/(x – 1)² 两者缺一不可；漏掉第一项是常见错误。将等式两边乘以完整分母后，综合运用代入法和系数比较法求解 A、B、C。

To verify: substituting x = 1 eliminates A and C, yielding B directly. Then substituting x = -3 yields C. Finally, comparing x² coefficients or substituting any convenient value gives A.

验证方法：代入 x = 1 消去 A 和 C，直接得到 B。再代入 x = -3 得到 C。最后比较 x² 系数或代入任一方便的值求出 A。

4. Case 3: Irreducible Quadratic Factors | 情况三：不可约二次因式

When the denominator contains a quadratic factor with no real roots (discriminant Δ < 0), the corresponding partial fraction has a linear numerator. For example:

当分母含有无实数根的二次因式时（判别式 Δ < 0），对应的部分分式分子为线性式。例如：

(3x + 1) / [(x² + 1)(x – 2)] = (Ax + B)/(x² + 1) + C/(x – 2)

Thus, for each irreducible quadratic factor (ax² + bx + c)ⁿ, include terms of the form (Aₖx + Bₖ) / (ax² + bx + c)ᵏ for k = 1, 2, …, n.

因此，对于每个不可约二次因式 (ax² + bx + c)ⁿ，需包含形如 (Aₖx + Bₖ) / (ax² + bx + c)ᵏ 的项，其中 k = 1, 2, …, n。

Solving requires equating coefficients since substitution alone cannot eliminate all unknowns. This is because the quadratic factor contributes two unknowns per term.

由于每个二次因式项贡献两个未知数，仅靠代入法无法消除所有未知数，因此需要联立比较系数求解。

5. The Cover-Up Method | 覆盖法技巧

The cover-up method provides a rapid way to find constants for distinct linear factors. To find the constant corresponding to factor (x – a), cover the term (x – a) in the original denominator and evaluate the remaining expression at x = a.

覆盖法为求互异线性因式的常数提供了一种快捷方式。要求因子 (x – a) 对应的常数，只需在原分母中遮盖 (x – a) 项，并在 x = a 处计算剩余表达式的值。

For example, to decompose (4x + 1) / [(x – 2)(x + 3)]:

4x + 1 over (x – 2)(x + 3) = A/(x – 2) + B/(x + 3)

Cover (x – 2): evaluate (4x + 1)/(x + 3) at x = 2, giving A = 9/5. Cover (x + 3): evaluate (4x + 1)/(x – 2) at x = -3, giving B = 11/5.

遮盖 (x – 2)：在 x = 2 处计算 (4x + 1)/(x + 3)，得 A = 9/5。遮盖 (x + 3)：在 x = -3 处计算 (4x + 1)/(x – 2)，得 B = 11/5。

This method works only for distinct linear factors and can be combined with other techniques for more complex denominators.

该方法仅适用于互异线性因式，也可与其他技巧结合处理更复杂的分母。

6. Improper Fractions: Polynomial Division | 假分式：多项式除法

When the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is improper. The first step is to perform polynomial long division to obtain a polynomial quotient plus a proper remainder fraction.

当分子的次数大于或等于分母的次数时，该分式为假分式。第一步需进行多项式长除法，得到多项式商加上一个真分式余项。

(x³ + 2x² – 3x + 1) / (x² – 1) = x + 2 + (x + 3) / (x² – 1)

Only the proper remainder fraction then undergoes partial fraction decomposition. The polynomial part remains as is and represents the asymptotic behaviour of the original function.

只有真分式余项需要进一步部分分式分解。多项式部分保持不变，它反映了原函数的渐近行为。

This division step is essential before applying integration or expansion techniques, as those methods typically require proper rational functions.

在应用积分或展开技巧之前必须完成此除法步骤，因为这些方法通常要求真分式形式。

7. Application in Integration | 在积分中的应用

Partial fraction decomposition transforms complex rational integrals into a sum of simpler integrals, each solvable by standard formulas. Consider:

部分分式分解将复杂的有理函数积分转化为若干简单积分之和，每个简单积分都可通过标准公式求解。考虑：

∫ [4/(x – 1) + 3/(x + 2)] dx = 4·ln|x – 1| + 3·ln|x + 2| + C

Each term 1/(x – a) integrates to ln|x – a|, while terms with quadratic denominators integrate to arctan or logarithmic forms. For repeated factors, the integral of 1/(x – a)ⁿ for n ≥ 2 uses the power rule.

每个 1/(x – a) 形式的项积分后得到 ln|x – a|，而含二次分母的项积分后得到 arctan 或对数形式。对于重复因式，n ≥ 2 时 1/(x – a)ⁿ 的积分使用幂法则。

This application is one of the most frequently examined topics in A-Level paper questions, often combined with determining the constant of integration from boundary conditions.

该应用是 A-Level 考题中最常见的考点之一，常与利用边界条件确定积分常数相结合。

8. Application in Binomial Expansion | 在二项展开中的应用

Partial fractions enable the binomial expansion of rational functions when x is small. After decomposition, each partial fraction can be rewritten in the form k(1 ± ax)ⁿ and expanded using the binomial series.

部分分式分解使我们能在 x 较小时对有理函数进行二项展开。分解后，每个部分分式可改写为 k(1 ± ax)ⁿ 的形式，并利用二项级数展开。

1/[(1 – x)(1 + 2x)] = 1/(1 – x) – 2/(1 + 2x)

Each term expands as a geometric or binomial series:

1/(1 – x) = 1 + x + x² + x³ + …

-2/(1 + 2x) = -2[1 – 2x + 4x² – 8x³ + …]

Adding the series gives the expansion valid for |x| < 1/2, the intersection of both convergence intervals. Fractional and negative powers require the generalised binomial theorem.

将两个级数相加即得到 |x| < 1/2 范围内有效的展开式，这是两个收敛区间的交集。分数次幂和负次幂需要使用广义二项式定理。

9. Systematic Procedure | 系统化解题流程

A reliable step-by-step approach can be summarised as follows:

可靠的逐步解题流程可总结如下：

Step 1: Check if the fraction is proper; if not, perform polynomial division.

第一步：检查是否为真分式；若不是，先进行多项式除法。
Step 2: Factor the denominator completely into linear and irreducible quadratic factors.

第二步：将分母完全分解为线性因式和不可约二次因式。
Step 3: Write the expected partial fraction form based on the factor types.

第三步：根据因式类型写出预期的部分分式形式。
Step 4: Multiply through by the original denominator to eliminate fractions.

第四步：两边乘以原分母以消去分式。
Step 5: Solve for unknown constants using substitution and/or coefficient comparison.

第五步：运用代入法和/或系数比较法求解未知常数。
Step 6: Verify the result by substituting a test value or recombining the terms.

第六步：代入测试值或将各项重新合并以验证结果。

Following this procedure systematically reduces errors. Always write the general decomposition form before computing constants.

按此流程系统化操作可减少错误。在计算常数之前，务必先写出通用分解形式。

10. Common Pitfalls | 常见错误与对策

Students frequently make mistakes in the following areas. Avoiding these errors is key to scoring full marks.

学生常在以下几个方面出错。避免这些错误是获得满分的关键。

Mistake \| 错误	Correction \| 纠正
Omitting terms for repeated factors	Include all powers from 1 to n
漏掉重复因式的项	包含从 1 到 n 的所有幂次
Forgetting polynomial division for improper fractions	Always check degrees first
假分式忘记多项式除法	先检查分子分母次数
Wrong numerator for quadratic factors	Use linear numerator Ax + B
二次因式对应分子写错	应使用线性分子 Ax + B
Sign errors when substituting negative values	Carefully track signs in brackets
代入负值时符号出错	仔细跟踪括号内的符号

Additionally, always check that the number of unknown constants equals the degree of the denominator. If not, return to step 3 and re-examine the proposed form.

此外，始终检查未知常数的数量是否等于分母的次数。若不相等，请返回第三步重新检查所设的形式。

11. Worked Example | 完整例题

Consider the decomposition of (2x² + 5x + 7) / [(x + 1)(x² + 4)]. First, note this is a proper fraction. The irreducible quadratic factor x² + 4 requires a linear numerator:

考虑分解 (2x² + 5x + 7) / [(x + 1)(x² + 4)]。首先，这是一个真分式。不可约二次因式 x² + 4 需要线性分子：

(2x² + 5x + 7) / [(x + 1)(x² + 4)] = A/(x + 1) + (Bx + C)/(x² + 4)

Multiplying through by the denominator:

2x² + 5x + 7 = A(x² + 4) + (Bx + C)(x + 1)

两边乘以分母：

2x² + 5x + 7 = A(x² + 4) + (Bx + C)(x + 1)

Substitute x = -1: 2 – 5 + 7 = A(1 + 4) ⇒ 4 = 5A ⇒ A = 4/5. Expanding and comparing coefficients:

代入 x = -1：2 – 5 + 7 = A(1 + 4) ⇒ 4 = 5A ⇒ A = 4/5。展开并比较系数：

A + B = 2 ⇒ B = 2 – 4/5 = 6/5

B + C = 5 ⇒ C = 5 – 6/5 = 19/5

The final decomposition is therefore:

2x² + 5x + 7 over (x + 1)(x² + 4) = 4/[5(x + 1)] + (6x + 19)/[5(x² + 4)]

因此最终分解为：

2x² + 5x + 7 除以 (x + 1)(x² + 4) = 4/[5(x + 1)] + (6x + 19)/[5(x² + 4)]

12. Exam Strategies and Summary | 应试策略与总结

In the exam, partial fraction questions typically appear in Pure Mathematics papers, either standalone or embedded within integration or series expansion problems. Start by identifying the factor structure of the denominator, then select the appropriate decomposition template.

考试中，部分分式问题通常出现在纯数学卷中，要么独立设问，要么嵌入积分或级数展开问题中。首先识别分母的因式结构，然后选择相应的分解模板。

Key takeaways to remember:

需要牢记的要点：

Distinct linear factors: one constant per factor, solvable by substitution.

互异线性因式：每个因式一个常数，可用代入法求解。
Repeated linear factors: include all ascending powers, use coefficient comparison.

重复线性因式：包含所有升幂项，使用系数比较法。
Irreducible quadratics: use linear numerators, combine methods to solve.

不可约二次因式：使用线性分子，综合方法求解。
Improper fractions: perform polynomial division first, then decompose the remainder.

假分式：先进行多项式除法，再分解余项。

With consistent practice, partial fraction decomposition becomes a reliable, high-scoring tool in your mathematical toolkit.

通过持续练习，部分分式分解将成为一个可靠且高得分的数学工具。

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A-Level Mathematics: Partial Fraction Decomposition Methods | A-Level 数学：部分分式分解方法