Partial Fraction Decomposition Calculator
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How to Decompose Partial Fractions
For linear factors:
(ax + b)/((x + c)(x + d)) = A/(x + c) + B/(x + d)
For quadratic factors:
(ax + b)/(x² + cx + d) = (Ax + B)/(x² + cx + d)
Example Problems
Example 1: Linear Factors
Decompose: (2x + 3)/((x + 1)(x + 2))
Step 1: Set up the decomposition
(2x + 3)/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2)
Step 2: Solve for A and B
A = 1, B = 1
Result: 1/(x + 1) + 1/(x + 2)
Example 2: Quadratic Factor
Decompose: (x + 2)/(x² + 3x + 2)
Step 1: Factor the denominator
(x + 2)/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2)
Step 2: Solve for A and B
A = 1, B = 0
Result: 1/(x + 1)
Common Cases
| Case | Decomposition |
|---|---|
| Distinct Linear Factors | P(x)/((x-a)(x-b)) = A/(x-a) + B/(x-b) |
| Repeated Linear Factor | P(x)/(x-a)² = A/(x-a) + B/(x-a)² |
| Quadratic Factor | P(x)/(x²+ax+b) = (Ax+B)/(x²+ax+b) |
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