A-Level Maths: Partial Fractions | A-Level 数学：部分分式

📚 A-Level Maths: Partial Fractions | A-Level 数学：部分分式

Partial fractions is a method of rewriting a rational expression as a sum of simpler fractions. It is an essential skill in A-Level Mathematics, particularly for integration and series expansion.

部分分式是一种将有理表达式改写为若干更简单分式之和的方法。它是 A-Level 数学中的核心技巧，尤其在积分和级数展开中非常重要。

1. Proper and Improper Rational Fractions | 真分式与假分式

A rational expression is the quotient of two polynomials. It is proper when the degree of the numerator is less than the degree of the denominator. For example, (3x + 2)/(x² + x + 1) is proper. If the numerator has degree equal to or greater than the denominator, the expression is improper, and polynomial division must be performed before decomposition.

有理表达式是两个多项式之比。当分子的次数小于分母的次数时，称为真分式。例如，(3x + 2)/(x² + x + 1) 是真分式。如果分子的次数大于或等于分母的次数，则它是假分式，必须先进行多项式除法，再作分解。

Before using partial fractions, always check whether the expression is proper. If it is improper, divide the numerator by the denominator to obtain a polynomial plus a proper fraction.

在使用部分分式前，务必检查表达式是否为真分式。若是假分式，先用分子除以分母，得到一个多项式加上一个真分式。

2. Distinct Linear Factors | 互异线性因子

Suppose the denominator is a product of distinct linear factors. For each factor (ax + b), we include a term A/(ax + b).

若分母是一组互不相同的线性因子的乘积，则对每个因子 (ax + b)，我们加入一项 A/(ax + b)。

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

Multiply both sides by the denominator (x − 1)(x + 2) to obtain:

两边同乘分母 (x − 1)(x + 2)，得到：

x + 3 = A(x + 2) + B(x − 1)

Substitute x = 1 to eliminate the B term: 4 = 3A, so A = 4/3. Substitute x = −2 to eliminate the A term: 1 = −3B, so B = −1/3.

代入 x

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