Partial Fractions | 部分分式

📚 Partial Fractions | 部分分式

In Edexcel A-Level Mathematics (Pure Mathematics), partial fractions are a core algebraic technique used to break a complicated rational expression into a sum of simpler fractions. They are essential for integration, binomial expansion, and solving certain types of equations. This guide covers the main forms required by the specification, including linear factors, repeated linear factors and irreducible quadratic factors.

在爱德思 A-Level 数学（纯数）中，部分分式是一项核心代数技巧，用于把一个复杂的有理式拆分成若干个较简单的分式之和。它是积分、二项式展开以及求解某些方程的重要工具。本指南涵盖 Edexcel 考纲要求的主要形式，包括线性因式、重复线性因式和不可约二次因式。

1. What Are Partial Fractions? | 什么是部分分式？

A rational function is the ratio of two polynomials. When the degree of the numerator is less than the degree of the denominator, the rational function is called proper. Partial fractions rewrite a proper rational function as a sum of simpler fractions whose denominators are linear or quadratic factors.

有理函数是两个多项式的比值。当分子的次数小于分母的次数时，该有理函数称为真分式。部分分式把一个真分式改写成若干个分母为线性因式或二次因式的简单分式之和。

(5x + 1) / ((x + 1)(x − 2)) = A / (x + 1) + B / (x − 2)

The constants A and B must be found so that the two sides are identical for all values of x within the domain.

常数 A 和 B 需要被求出，使得等式两边在定义域内对所有 x 都恒等。

2. Proper and Improper Rational Functions | 真分式与假分式

Before splitting a rational function, always check the degrees of the numerator and denominator. If the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is improper and must be divided first using polynomial long division. The remainder term is then split into partial fractions.

在拆分有理函数之前，一定要先检查分子和分母的次数。如果分子的次数大于或等于分母的次数，该分式就是假分式，必须先使用多项式长除法进行化简。余项再进行部分分式分解。

Proper: degree of numerator < degree of denominator.
Improper: degree of numerator ≥ degree of denominator — divide first.
After division, write the remainder over the original denominator and split only that proper fraction.

真分式：分子次数 < 分母次数。假分式：分子次数 ≥ 分母次数——先做长除法。除法后，将余项除以原分母，只对这个真分式进行拆分。

3. Linear Factors and the Cover-Up Method | 线性因式与覆盖法

When the denominator has distinct linear factors, such as (x + 1)(x − 2), write the expression as A/(x + 1) + B/(x − 2). To find A, multiply both sides by (x + 1) and substitute x = −1. To find B, multiply by (x − 2) and substitute x = 2. This shortcut is often called the cover-up method.

当分母具有不同的线性因式时，例如 (x + 1)(x − 2)，可将表达式写成 A/(x + 1) + B/(x − 2)。求 A 时，两边乘以 (x + 1)，再代入 x = −1。求 B 时，两边乘以 (x − 2)，再代入 x = 2。这种巧算方法通常称为覆盖法。

(3x + 5) / ((x + 1)(x − 2)) = A / (x + 1) + B / (x − 2)

Using the cover-up method:

使用覆盖法：

A = (3(−1) + 5) / (−1 − 2) = 2 / −3 = −2/3

B = (3(2) + 5) / (2 + 1) = 11 / 3

Therefore, the original fraction equals −2 / (3(x + 1)) + 11 / (3(x − 2)).

因此，原分式等于 −2 / (3(x + 1)) + 11 / (3(x − 2))。

4. Coefficient Comparison Method | 系数比较法

If substitution is messy or the denominator has factors with no real roots, compare coefficients of like powers of x after multiplying both sides by the common denominator. This gives simultaneous equations for the unknown constants.

如果代入法比较繁琐，或者分母含有无实数根的因式，可以在两边乘以公分母后，比较同次幂 x 的系数。这样可以得到关于未知常数的联立方程组。

For example, suppose:

例如，假设：

(2x + 7) / ((x + 1)(x + 3)) = A / (x + 1) + B / (x + 3)

Multiplying by (x + 1)(x + 3) and expanding gives 2x + 7 = A(x + 3) + B(x + 1).

两边乘以 (x + 1)(x + 3) 并展开，得到 2x + 7 = A(x + 3) + B(x + 1)。

Equate coefficients of x: A + B = 2. Equate constant terms: 3A + B = 7.

比较 x 的系数：A + B = 2。比较常数项：3A + B = 7。

Solving gives A = 5/2 and B = −1/2.

解得 A = 5/2，B = −1/2。

5. Repeated Linear Factors | 重复线性因式

For a denominator containing a repeated factor such as (x − 2)², the decomposition must include one term for every power from 1 up to the repeated power. That means A/(x − 2) + B/(x − 2)², not just A/(x − 2).

当分母包含重复因式（例如 (x − 2)²）时，分解必须包含从 1 次幂到重复最高次幂的每一项。也就是说，要写成 A/(x − 2) + B/(x − 2)²，而不能只是 A/(x − 2)。

(4x + 1) / ((x − 2)²(x + 1)) = A/(x − 2) + B/(x − 2)² + C/(x + 1)

You can find B and C quickly using the cover-up method: B is obtained by multiplying by (x − 2)² and substituting x = 2; C is obtained by multiplying by (x + 1) and substituting x = −1. The remaining constant A is then found by comparing coefficients or substituting another simple value of x.

B 和 C 可以用覆盖法快速求出：B 通过乘以 (x − 2)² 并代入 x = 2 得到；C 通过乘以 (x + 1) 并代入 x = −1 得到。剩下的常数 A 可以通过比较系数或代入另一个简单的 x 值求出。

6. Irreducible Quadratic Factors | 不可约二次因式

When the denominator has an irreducible quadratic factor such as x² + 1 or x² + x + 1, the numerator of the corresponding partial fraction must be linear, not constant. The general form is (Ax + B)/(x² + 1). This is because the numerator should have degree one less than the denominator.

当分母含有不可约二次因式（例如 x² + 1 或 x² + x + 1）时，对应部分分式的分子必须是线性的，而不是常数。一般形式为 (Ax + B)/(x² + 1)。

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Partial Fractions | 部分分式