## Partial Fractions

An expression of the form \(\frac { f(x) }{ g(x) }\), where f(x) and g(x) are polynomial in x, is called a rational fraction.

**Proper rational functions:**Functions of the form \(\frac { f(x) }{ g(x) }\), where f(x) and g(x) are polynomials and g(x) ≠ 0, are called rational functions of x.

If degree of f(x) is less than degree of g(x),then is called a proper rational function.**Improper rational functions:**If degree of f(x) is greater than or equal to degree of g(x), then \(\frac { f(x) }{ g(x) }\) is called an improper rational function.**Partial fractions:**Any proper rational function can be broken up into a group of different rational fractions, each having a simple factor of the denominator of the original rational function. Each such fraction is called a partial fraction.

If by some process, we can break a given rational function \(\frac { f(x) }{ g(x) }\) into different fractions, whose denominators are the factors of g(x),then the process of obtaining them is called the resolution or decomposition of \(\frac { f(x) }{ g(x) }\) into its partial fractions.

### Different cases of partial fractions

**(1) When the denominator consists of non-repeated linear factors:**

To each linear factor (x – a) occurring once in the denominator of a proper fraction, there corresponds a single partial fraction of the form \(\frac { A }{ x-a }\), where A is a constant to be determined.

If g(x) = (x – a_{1})(x – a_{2})(x – a_{3}) ……. (x – a_{n}), then we assume that,

where A_{1}, A_{2}, A_{3}, ………. A_{n }are constants, can be determined by equating the numerator of L.H.S. to the numerator of R.H.S. (after L.C.M.) and substituting x = a_{1}, a_{2},…… a_{n}.

**(2) When the denominator consists of linear factors, some repeated:**

To each linear factor (x – a) occurring r times in the denominator of a proper rational function, there corresponds a sum of r partial fractions.

Let g(x) = (x – a)^{k}(x – a_{1})(x – a_{2}) ……. (x – a_{r}). Then we assume that

Where A_{1}, A_{2}, A_{3}, ………. A_{k }are constants. To determine the value of constants adopt the procedure as above.

**(3) When the denominator consists of non-repeated quadratic factors:**

To each irreducible non repeated quadratic factor ax^{2} + bx + c, there corresponds a partial fraction of the form \(\frac { Ax+B }{ { a{ x }^{ 2 }+bx+c } }\), where A and B are constants to be determined.

Example :

**(4) When the denominator consists of repeated quadratic factors:**

To each irreducible quadratic factor ax^{2} + bx + c occurring r times in the denominator of a proper rational fraction there corresponds a sum of r partial fractions of the form.

where, A’s and B’s are constants to be determined.

### Partial fractions of improper rational functions

If degree of is greater than or equal to degree of g(x), then \(\frac { f(x) }{ g(x) }\) is called an **improper rational function** and every rational function can be transformed to a proper rational function by dividing the numerator by the denominator.

We divide the numerator by denominator until a remainder is obtained which is of lower degree than the denominator.

### General method of finding out the constants

- Express the given fraction into its partial fractions in accordance with the rules written above.
- Then multiply both sides by the denominator of the given fraction and you will get an identity which will hold for all values of x.
- Equate the coefficients of like powers of x in the resulting identity and solve the equations so obtained simultaneously to find the various constant is short method. Sometimes, we substitute particular values of the variable x in the identity obtained after clearing of fractions to find some or all the constants. For non-repeated linear factors, the values of x used as those for which the denominator of the corresponding partial fractions become zero.