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Partial fraction decomposition of Proper rational expressions that consist of Non repeated linear factors

Formula

$\dfrac{f(x)}{(x-a_1)(x-a_2)(x-a_3) \ldots (x-a_n)}$ $\,=\,$ $\dfrac{C_1}{x-a_1}$ $+$ $\dfrac{C_2}{x-a_2}$ $+$ $\dfrac{C_3}{x-a_3}$ $+$ $\cdots$ $+$ $\dfrac{C_n}{x-a_n}$

Introduction

In some cases, the rational expressions that consist of the following two properties should have to be decomposed into partial fractions in mathematics.

  1. The degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
  2. The polynomial in the denominator can be expressible as the product of non-repeating linear factors.

Now, let us understand the partial fractions decomposition when the denominator of the proper rational function can be expressible as non-repeated linear factors.

Let $f(x)$ and $g(x)$ be polynomials. The quotient of $f(x)$ by $g(x)$ is a rational function, which is denoted by $r(x)$. The relation between the three functions are written as follows.

$r(x) \,=\, \dfrac{f(x)}{g(x)}$

Let us assume that the degree of function $f(x)$ is less than the degree of the function $g(x)$. Therefore, the rational expression $r(x)$ is a proper function in rational form. Now, assume that the polynomial in the denominator is expressible as a product of non-repeated linear factors.

$g(x)$ $\,=\,$ $(x-a_1)(x-a_2)(x-a_3) \ldots (x-a_n)$

In this case, $x-a_1$, $x-a_2$, $x-a_3$, $\ldots$ $x-a_n$ are linear factors. $a_1$, $a_2$, $a_3$ $\ldots$ $a_n$ are the distinct constants. Now, the proper rational function can be written in the following mathematical form.

$\implies$ $r(x)$ $\,=\,$ $\dfrac{f(x)}{(x-a_1)(x-a_2)(x-a_3) \ldots (x-a_n)}$

This type of proper rational expression can be decomposed as the sum of the partial fractions.

$\implies$ $\dfrac{f(x)}{(x-a_1)(x-a_2)(x-a_3) \ldots (x-a_n)}$ $\,=\,$ $\dfrac{C_1}{x-a_1}$ $+$ $\dfrac{C_2}{x-a_2}$ $+$ $\dfrac{C_3}{x-a_3}$ $+$ $\cdots$ $+$ $\dfrac{C_n}{x-a_n}$

In this case, $C_1$, $C_2$, $C_3$ $\ldots$ $C_n$ are constants.

Example

$\dfrac{3x}{(x+1)(x-2)}$ $\,=\,$ $\dfrac{1}{x+1}$ $+$ $\dfrac{2}{x-2}$

In this example, $\dfrac{3x}{(x+1)(x-2)}$ is a rational function. $3x$ and $(x+1)(x-2)$ are polynomials in the numerator and denominator. If you observe that the degree of the function $3x$ is less than the degree of the polynomial $(x+1)(x-2)$. Hence, the rational function $\dfrac{3x}{(x+1)(x-2)}$ is also called as the proper rational expression.

The proper rational function that has non-repeated linear factors is decomposed as the sum of the partial fractions $\dfrac{1}{x+1}$ and $\dfrac{2}{x-2}$.

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