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# Partial Fractions

A fraction is a number expressed in form of $\frac{a}{b}$, where the numerator is 'a' and the denominator is 'b'. Fractions are of two kinds: like fractions and unlike fractions. If the denominators of the two fractions are same, then it is a like fractions. If the denominators of the two fractions are not same, then the fractions are unlike.

Apart from like and unlike fractions there are also different other types of fractions. If the numerator value is less than the denominator value, then the fraction is a proper fraction. If the numerator value is greater than the denominator value, then the fraction is a improper fraction.

All mathematical operations namely addition, subtraction, multiplication and division can be performed on the fractions. Fractions can also be compare to find the greatest and smallest among fractions. Thus a series of fraction Numbers can be arranged in ascending or descending order.  We can also convert the given fraction into Percentage or decimal numbers.

Partial Fractions are a form of fractions that have complex equations, both in numerator and the denominator.  These fractions are solved by using the technique of decomposition which is an algebraic method to make a complex Rational Function (a fraction) into a simpler one. Solving Partial Fractions can be explained by an example:

Example:
Solve  $\frac{x^{2} + 2x - 1}{2x^{3} + 3x^{2} - 2x}$
Solve it by factoring the denominator. We get,
$\frac{x^{2} + 2x - 1}{2x^{3} + 3x^{2} - 2x}$ = $\frac{x^{2} + 2x - 1}{x (2x - 1) (x + 2)}$
$\frac{x^{2} + 2x - 1}{2x^{3} + 3x^{2} - 2x}$ = $\frac{A}{ x}$ + $\frac{B}{2x - 1}$ + $\frac{C}{x + 2}$
x2 + 2x - 1 = A (2x - 1)(x + 2) + B (x + 2)x + C (2x - 1)x
Substituting x = 0 ; x = x = $\frac{1}{2}$ ; x = -2 we get the value of A, B and C respectively.
A = $\frac{1}{2}$
B = $\frac{1}{5}$
C = $\frac{-1}{10}$
Now our equation can be written as,
$\frac{x^{2} + 2x - 1}{2x^{3} + 3x^{2} - 2x}$ = $\frac{\frac{1}{2}}{ x}$ + $\frac{\frac{1}{5}}{2x - 1}$ + $\frac{\frac{-1}{10}}{x + 2}$

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