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Partial fraction decomposition of Repeated Linear factors

Formula

$\dfrac{f(x)}{(ax+b)^{\displaystyle n}}$ $\,=\,$ $\dfrac{C_1}{ax+b}$ $+$ $\dfrac{C_2}{(ax+b)^2}$ $+$ $\dfrac{C_3}{(ax+b)^3}$ $+$ $\cdots$ $+$ $\dfrac{C_{\displaystyle n}}{(ax+b)^{\displaystyle n}}$

Decomposing a proper rational function that consists of the repeated linear factors in the denominator, into partial fractions is called the partial fraction decomposition of the repeated linear factors.

Introduction

Let $f(x)$ and $g(x)$ be two polynomials. The ratio of the function $f(x)$ to function $g(x)$ forms a rational function. Suppose, the degree of the polynomial $g(x)$ is greater than the degree of the polynomial $f(x)$. Then, the rational function is called a proper rational function.

$\dfrac{f(x)}{g(x)}$

In this case, the denominator of proper rational function is in repeated linear factor form. In other words, if the linear expression in one variable is represented by $ax+b$, and is multiplied by itself $n$ times, then the product of them is the function in the denominator. It means, $g(x) = (ax+b)^{\displaystyle n}$. Now, the proper rational expression is written as follows.

$\implies$ $\dfrac{f(x)}{g(x)}$ $\,=\,$ $\dfrac{f(x)}{\underbrace{(ax+b) \times (ax+b) \times \cdots \times (ax+b)}_{\displaystyle n \, linear \, factors}}$

$\,\,\,\therefore \,\,\,\,\,\,$ $\dfrac{f(x)}{g(x)}$ $\,=\,$ $\dfrac{f(x)}{(ax+b)^{\displaystyle n}}$

This rational function is called the proper rational function with repeated linear factors. It can be decomposed as partial fractions in the following form.

$\dfrac{f(x)}{(ax+b)^{\displaystyle n}}$ $\,=\,$ $\dfrac{C_1}{ax+b}$ $+$ $\dfrac{C_2}{(ax+b)^2}$ $+$ $\dfrac{C_3}{(ax+b)^3}$ $+$ $\cdots$ $+$ $\dfrac{C_{\displaystyle n}}{(ax+b)^{\displaystyle n}}$

In this case, $C_1,$ $C_2,$ $C_3,$ $\cdots$ $C_{\displaystyle n}$ are the real numbers.

Problems

The list of questions on partial fraction decomposition with solutions to learn how to decompose the proper rational function with repeated linear factors into partial fractions.

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