Partial Fraction Roots Calculator

What is Partial Fraction Decomposition?

Partial fraction decomposition is a method in algebra that allows us to break down complex rational expressions into simpler fractions. This technique is particularly useful when solving differential equations, integrating functions, and analyzing systems of linear equations.

The general form of a partial fraction decomposition is:

Where f(x) and g(x) are polynomials, and the right side consists of simpler fractions with denominators that are factors of g(x).

How to Use the Partial Fraction Roots Calculator

Our calculator makes it easy to perform partial fraction decomposition and find the roots of the resulting equations. Here's how to use it:

1. Enter the numerator and denominator of your rational expression in the input fields.
2. Select the type of roots you want to find (real, complex, or all).
3. Click "Calculate" to perform the decomposition and find the roots.
4. Review the results, which will show the partial fractions and their roots.

The Formula Explained

The partial fraction decomposition process involves solving for the constants A, B, etc. in the equation:

This is typically done by multiplying both sides by the denominator and then equating coefficients. The roots of the resulting equations can then be found using the quadratic formula for quadratic factors.

Worked Example

Let's solve the equation (x² + 3x + 2)/(x² + 2x + 1).

1. Factor the denominator: x² + 2x + 1 = (x + 1)²
2. Assume the partial fraction form: (x² + 3x + 2)/[(x + 1)²] = A/(x + 1) + B/(x + 1)²
3. Multiply both sides by (x + 1)²: x² + 3x + 2 = A(x + 1) + B
4. Expand and solve for A and B: A = 2, B = 1
5. The partial fractions are: 2/(x + 1) + 1/(x + 1)²
6. The roots of the denominator are x = -1 (double root)