How to Use the Partial Fraction Decomposition Calculator
Partial fraction decomposition is a technique used to break down complex rational expressions into a sum of simpler fractions. This partial fraction decomposition calculator helps students and engineers quickly find the constants required for integration, Laplace transforms, and series expansions.
- Numerator Coefficients
- Enter the coefficients for the top of the fraction. For a linear numerator (Mx + N), enter M and N. For quadratic (Lx² + Mx + N), include L.
- Denominator Roots
- Instead of the full polynomial, enter the roots (a, b, c) from the factored form (x – a)(x – b). For example, for (x – 2), the root is 2. For (x + 3), the root is -3.
- Decomposition Type
- Choose between two or three distinct linear factors based on your denominator's degree.
How Partial Fraction Decomposition Works
When the degree of the numerator is less than the degree of the denominator (proper fraction), we can express the fraction as a sum of parts. The most common form involves distinct linear factors:
P(x) / ((x – a)(x – b)) = A / (x – a) + B / (x – b)
The Heaviside Cover-up Method is the fastest way to solve for the constants A and B:
- To find A: Multiply the whole equation by (x – a) and set x = a.
- To find B: Multiply the whole equation by (x – b) and set x = b.
- Result: Substitute the found constants back into the decomposed form.
Calculation Example
Example: Decompose (x + 5) / ((x – 2)(x + 3)) into partial fractions.
Step-by-step solution:
- Identify Numerator: M = 1, N = 5
- Identify Denominator Roots: a = 2, b = -3
- Solve for A (at x = 2): A = (2 + 5) / (2 – (-3)) = 7 / 5 = 1.4
- Solve for B (at x = -3): B = (-3 + 5) / (-3 – 2) = 2 / -5 = -0.4
- Final Result: 1.4 / (x – 2) – 0.4 / (x + 3)
Common Questions
When can I not use this calculator?
This calculator is designed for distinct linear factors. If your denominator contains irreducible quadratics (like x² + 1) or repeated roots (like (x – 1)²), the algebraic form changes slightly, requiring different solving techniques like system of equations.
What if the numerator degree is higher?
If the numerator's degree is equal to or higher than the denominator's, you must first perform polynomial long division. The remainder is then decomposed using the partial fraction method.
Is this useful for Calculus?
Yes! Partial fraction decomposition is a critical step in finding the integrals of rational functions. It transforms a difficult product into a sum of natural logarithms (ln), which are much easier to integrate.