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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.
Formula
Express rational function as sum of simpler fractions: P(x)/Q(x) = A/(x−a) + B/(x−b) + ...
- P(x)
- numerator polynomial
- Q(x)
- denominator polynomial
- A, B, ...
- coefficients of partial fractions
Guida passo passo
- 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
Esempi risolti
Ingresso
(3x+5)/((x+1)(x+2))
Risultato
A/(x+1) + B/(x+2); A=2, B=1
Ingresso
1/(x²−1)
Risultato
1/((x−1)(x+1)) = ½/(x−1) − ½/(x+1)
Domande frequenti
When is partial fractions useful?
Integration: ∫ P(x)/Q(x) dx becomes simpler. System solving and signal processing.
What if the numerator has degree ≥ denominator?
Use polynomial long division first. Then apply partial fractions to the remainder.
How do I handle repeated roots?
For root r repeated k times: include A/(x−r) + B/(x−r)² + ... + Z/(x−r)ᵏ.
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