# 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)/(2x^{2}-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} |
A_{1}/(ax + b) + A_{2}/(ax + b)^{2} + .......... A_{n}/(ax + b)^{n} |

Non-repeated Quadratic Factor | (px^{2} + qx + r)/(ax^{2} + bx + c) |
(Ax + B)/(ax^{2} + bx + c) |

Repeated Quadratic Factor | (px^{2} + qx + r)/(ax^{2} + bx + c)^{n} |
(A_{1}x + B_{1})/(ax^{2} + bx + c) + (A_{2}x + B_{2})/(ax^{2} + bx + c)^{2} + ...(A_{n}x + B_{n})/(ax^{2} + 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)(x
^{2}+ 1)] = [A/(x - 2)] + [(Bx + C)/(x^{2}+ 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.