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

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

- The degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
- 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.

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

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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