Partial Fractions

What is a Partial Fraction?

An algebraic fraction can be broken down into simpler parts known as “partial fractions“. Consider an algebraic fraction, (3x+5)/(2x2-5x-3). This expression can be split into simple form like [2/(x – 3)] – [1/(2x + 1)].

The simpler parts [2/(x – 3)] and [1/(2x + 1)] are known as partial fractions.

Partial Fractions Formulas

In the above example, the numerators of partial fractions are 1 and 3. The numerator of a partial fraction is not always a constant. If the denominator is a linear function, the numerator is constant. And, if the denominator is a quadratic equation, then the numerator is linear. It means, the numerator's degree of a partial fraction is always one less than the denominator's degree. Further, the rational expression needs to be a proper fraction to be decomposed into a partial fraction. Listed below in the table are partial fraction formulas (here, all variables apart from x are constants).

Type Form of Rational Fraction Partial Fraction Decomposition
Non-repeated Linear Factor (px + q)/(ax + b) A/(ax + b)
Repeated Linear Factor (px + q)/(ax + b)n A1/(ax + b) + A2/(ax + b)2 + .......... An/(ax + b)n
Non-repeated Quadratic Factor (px2 + qx + r)/(ax2 + bx + c) (Ax + B)/(ax2 + bx + c)
Repeated Quadratic Factor (px2 + qx + r)/(ax2 + bx + c)n (A1x + B1)/(ax2 + bx + c) + (A2x + B2)/(ax2 + bx + c)2 + ...(Anx + Bn)/(ax2 + bx + c)n

Let us look at a few examples of partial fractions.

  • 4/[(x - 1)(x + 5)] = [A/(x - 1)] + [B/(x + 5)]
  • (3x + 1)/[(2x - 1)(x + 2)2] = [A/(2x - 1)] + [B/(x + 2)] + [C/(x + 2)2]
  • (2x - 3)/[(x - 2)(x2 + 1)] = [A/(x - 2)] + [(Bx + C)/(x2 + 1)]

In all these examples, A, B, and C are constants to be determined. Let's learn how to find these constants.

Partial Fraction Decomposition

The partial fraction decomposition is writing a rational expression as the sum of two or more partial fractions. The following steps are helpful to understand the process to decompose a fraction into 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.