Partial Fractions Solver
Resolve into Partial Fractions
Supports distinct/repeated linear factors (up to power 2) and the form of irreducible quadratic factors. Ensure degree of numerator is less than degree of denominator.
Step-by-step Solution:
Enter numerator and denominator, then click "Decompose Fraction".
Understanding Partial Fractions
Partial fraction decomposition is a technique used to rewrite a rational function as a sum of simpler fractions. This solver handles cases with:
- Distinct linear factors: $\frac{N(x)}{(a_1x+b_1)(a_2x+b_2)\cdots} = \frac{C_1}{a_1x+b_1} + \frac{C_2}{a_2x+b_2} + \cdots$
- Repeated linear factors (up to power of 2): $\frac{N(x)}{(ax+b)^2\cdots} = \frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \cdots$
- Irreducible quadratic factors: $\frac{N(x)}{(\dots)(ax^2+bx+c)(\dots)} = \dots + \frac{Ax+B}{ax^2+bx+c} + \dots$ (where $b^2-4ac < 0$)
- Repeated irreducible quadratic factors (form shown, e.g. power 2): $\frac{N(x)}{(ax^2+bx+c)^2\cdots} = \frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + \cdots$
The degree of $N(x)$ must be less than the degree of $D(x)$. The solver primarily uses the Heaviside method for linear factor coefficients; quadratic factor coefficients are shown in form but not fully solved by this method (requires system of equations).