Partial Fraction Decomposition Calculator

This method is used to rewrite a complex rational expression (a fraction of polynomials) as a sum of simpler fractions.

Conditions:

• The degree of the numerator must be less than the degree of the denominator. If not, you must perform long division first.

Steps:

1. Factor the Denominator: Completely factor the denominator into linear factors (like \(ax+b\)) or irreducible quadratic factors (like \(ax^2+bx+c\)).
2. Set up the Form: For each factor in the denominator, create a term in the decomposition:
   • For each distinct linear factor \((ax+b)\), you get a term \(\frac{A}{ax+b}\).
   • For each repeated linear factor \((ax+b)^n\), you get terms \(\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \dots + \frac{A_n}{(ax+b)^n}\).
   • For each irreducible quadratic factor \((ax^2+bx+c)\), you get a term \(\frac{Ax+B}{ax^2+bx+c}\).
3. Solve for Coefficients: Multiply both sides by the original denominator and solve for the unknown coefficients (A, B, C, etc.) by equating coefficients of like terms.