Solve Partial Fraction Decomposition

Partial Fraction Decomposition:

\[ \frac{2x+3}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1} \] \[ A(x+1) + B(x-1) = 2x + 3 \]

Coefficient A:

Coefficient B:

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1. What is Partial Fraction Decomposition?

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are easier to work with, especially in calculus and differential equations.

2. How Does the Calculator Work?

The calculator solves equations of the form:

\[ \frac{mx + c}{(x - r_1)(x + r_2)} = \frac{A}{x - r_1} + \frac{B}{x + r_2} \]

Where:

\( m \) — Coefficient of x in the numerator
\( c \) — Constant term in the numerator
\( r_1, r_2 \) — Roots from the denominator
\( A, B \) — Coefficients to be found

Explanation: The calculator solves the system of equations derived by equating coefficients from both sides of the equation.

3. Importance of Partial Fractions

Details: Partial fractions are essential for integration, Laplace transforms, and solving differential equations in engineering and physics.

4. Using the Calculator

Tips: Enter the coefficients of the numerator (mx + c) and the roots from the denominator (r₁ and r₂). The calculator will solve for A and B in the partial fraction decomposition.

5. Frequently Asked Questions (FAQ)

Q1: What if my denominator has repeated roots?
A: This calculator handles distinct linear factors. For repeated roots, additional terms with higher powers are needed in the decomposition.

Q2: Can this handle quadratic factors in the denominator?
A: No, this calculator is designed for denominators that can be factored into distinct linear terms.

Q3: What if I get fractional coefficients?
A: This is normal. Partial fraction coefficients are often fractions.

Q4: Why is partial fraction decomposition useful?
A: It simplifies complex rational expressions, making them easier to integrate or transform.

Q5: Can I use this for improper fractions?
A: No, you must first perform polynomial division if the numerator's degree is equal to or greater than the denominator's.