What is a Partial Fraction?

Partial Fractions are used to solve a complex rational expression into two or simpler fractions. Usually the fractions with algebraic expressions are complex and hence a little difficult to solve. Thus, we use the partial fractions to split the fractions into multiple subfractions. A partial fraction is the reverse of the process of the addition of rational expressions. While the decomposition process the denominator is often an algebraic expression which is factorized to facilitate the process of generating partial fractions.

We can simply do some arithmetic operations on an algebraic fraction to get a single rational expression. In reverse, this rational expression can be splitted into the reverse direction which involves the decomposition of partial fractions resulting in two partial fractions. Let us learn more about partial fractions in detail.

Definition of Partial Fractions

The process of splitting a rational expression in the form of two or more than two rational expressions is known as a partial fraction. The resultant rational expressions that are a part of the decomposition are called the partial fractions. This process is generally termed as splitting the given algebraic fraction into partial fractions. The denominator of the given algebraic expression has to be factorized to obtain the set of partial fractions.

Decomposition of Partial Fractions

As we have seen earlier, the decomposition of partial fractions is writing a rational expression as the sum of two or more partial fractions. The following steps are helpful to understand the process of decomposition of partial fractions.

  • Step-1: Factorize the numerator and denominator and simplify the rational expression, before doing partial fraction decomposition.
  • Step-2: Split the rational expression as per the formula for partial fractions. P/((ax + b)2 = [A/(ax + b)] + [B/(ax + b)2]. There are different partial fractions formulas based on the numerator and denominator expression.
  • Step-3: Take the LCM of the factors of the denominators of the partial fractions, and multiply both sides of the equation with this LCM.
  • Step-4: Simplify and obtain the values of A and B by comparing coefficients of like terms on both sides.
  • Step-5: Substitute the values of the constants A and B on the right side of the equation to obtain the partial fraction.

Partial Fractions of Improper Fraction

When we have to decompose an improper fraction into partial fractions, we first should do the long division. The long division is helpful to give a whole number and a proper fraction. The whole number is the quotient in the long division, and the remainder forms the numerator of the proper fraction, and the denominator is the divisor. The format of the result of the long division would be Quotient + Remainder/Divisor.

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