Partial Fractions - Resolving a given rational function into partial fractions

Partial fractions is a technique used to decompose a rational function into simpler fractions.
The rational function is a fraction where the numerator and the denominator are both polynomials.
The partial fraction decomposition involves breaking the rational function into a sum of simpler fractions with denominators that are linear factors or irreducible quadratic factors.
The technique involves finding the unknown coefficients in the partial fraction expression by equating the numerator of the rational function to the sum of the numerators of the partial fractions.
Partial fraction decomposition is used in integration of rational functions, solving differential equations, and signal processing.

To resolve a given rational function into partial fractions, first factorize the denominator of the rational function into linear or irreducible quadratic factors.
Express the partial fraction decomposition in the form of unknown coefficients over linear or irreducible quadratic factors.
Equate the numerator of the rational function to the sum of the numerators of the partial fractions, and solve for the unknown coefficients using algebraic methods.
If there are repeated factors in the denominator, then use the denominators raised to increasing powers as denominators in the partial fraction decomposition.
If the degree of the numerator is greater than or equal to the degree of the denominator, then perform long division to obtain a proper fraction, and then resolve the proper fraction into partial fractions.

Partial Fractions:

Let $R(x)$ and $Q(x)$ be two polynomials such that $deg(R(x)) < deg(Q(x))$. Then $R(x)/Q(x)$ can be expressed as a sum of partial fractions of the form: $$\\frac{A}{x-a} + \\frac{B}{(x-a)^2} + \\cdots + \\frac{L}{(x-a)^k} + \\frac{R(x)}{Q(x)}$$ where $a$ is a root of $Q(x)$ of multiplicity $k$ and $A, B, \\ldots, L$ are constants.

Resolving a Given Rational Function into Partial Fractions:

Let $R(x)$ and $Q(x)$ be two polynomials such that $deg(R(x)) < deg(Q(x))$. Then $R(x)/Q(x)$ can be expressed as a sum of partial fractions by following the steps given in the algorithm.