Partial Fractions: Introduction


Partial Fraction: If a rational fraction is decomposite into two or more easy fractions with simpler denominators than it is called decomposition of fraction or expansion of fraction and the expanded easy fractions are called partial fractions of the main fraction.

Rational Fraction: The rational fraction is an algebraic fraction with a polynomial nominator and denominator.

Example: Find the partial fractions of the given equation? $$ \frac{3x - 2}{(x - 1) \ (x - 2)} $$

Solution: The given equation can be expanded into partial fractions as given $$ \frac{3x - 2}{(x - 1) \ (x - 2)} = \frac{A}{(x - 1)} + \frac{B}{(x - 2)} $$ Here \(\frac{A}{(x - 1)} + \frac{B}{(x - 2)}\) are the partial fractions of the original equation.

Note: A given fraction can be expanded into partial fractions if and only if the given fraction is a "proper fraction".

Proper Fraction: If the degree of nominator is less than the degree of denominator then the fraction is called a proper fraction.

Example: \(\frac{x - 1}{x^2 - 2}\) Here the degree of nominator is 1 and the degree of the denominator is 2, so it is clear that the degree of nominator is less than the degree of the denominator. Hence it is a proper fraction.

Improper Fraction: If the degree of nominator is greater or equal to the degree of denominator then the fraction is called an improper fraction.

Example: \(\frac{2x^2 + 3}{x^2 + 1}\) Here the degree of nominator and degree of the denominator is equal, Hence it is an improper fraction.

Note: If the given fraction in the question is an improper fraction then we have to make it a proper fraction first by dividing with the denominator of the same fraction.

Example: \(\frac{x^2 + 7x}{x^2 + 2x - 4}\)

Solution: The given fraction is an improper fraction as the degree of nominator and denominator is same so we can make it proper fraction by dividing nominator from the denominator and we will get $$ 1 + \frac{5x + 4}{x^2 + 2x - 4} $$ Here in the new fraction the degree of nominator is less than the degree of the denominator, hence it is a proper fraction.