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

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.

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.

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.

Latest Math Topics

Apr 18, 2022

Apr 14, 2022

Apr 05, 2022

Mar 18, 2022

Mar 05, 2022

Latest Math Problems

Apr 06, 2022

Mar 22, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved