Partial Fractions Decomposition
(ax + b) / ((x + p)(x + q))
a (numerator coefficient)
b (numerator constant)
p (first root factor)
q (second root factor)
✓Partial Fraction Decomposition
0.3333/(x+1) + 0.6667/(x+-2)
A
0.3333
B
0.6667
Partial fraction decomposition breaks a complex rational expression into simpler fractions. It is essential for integration in calculus and for solving differential equations using Laplace transforms.
- 1For (ax+b)/((x+p)(x+q)): write as A/(x+p) + B/(x+q)
- 2Multiply both sides by denominator
- 3Equate coefficients or substitute values
- 4Solve for A and B
(3x+5)/((x+1)(x+2))=A/(x+1) + B/(x+2); A=2, B=1
1/(x²−1)=1/((x−1)(x+1)) = ½/(x−1) − ½/(x+1)
| Denominator Type | Form | Example |
|---|---|---|
| Linear factors | A/(x+a) | 1/((x+1)(x+2)) |
| Repeated linear | A/(x+a) + B/(x+a)² | 1/(x+1)² |
| Irreducible quadratic | (Ax+B)/(x²+bx+c) | x/(x²+1) |
References
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