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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.
Công thức
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
Hướng dẫn từng bước
- 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
Ví dụ có lời giải
đầu vào
(3x+5)/((x+1)(x+2))
Kết quả
A/(x+1) + B/(x+2); A=2, B=1
đầu vào
1/(x²−1)
Kết quả
1/((x−1)(x+1)) = ½/(x−1) − ½/(x+1)
Câu hỏi thường gặp
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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