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

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.

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