how is partial fraction decomposition done

Partial fraction decomposition is the process of breaking down a rational function (a fraction with polynomial expressions in the numerator and denominator) into simpler fractions. The steps involved are:

1. Factor the denominator of the rational function into linear and quadratic factors.

2. Write the rational function as a sum of fractions where the numerator of each fraction is a constant or a polynomial of degree less than the degree of the denominator's factor.

3. Set up a system of equations by equating the numerators of the partial fraction decomposition to the original numerator.

4. Solve the system of equations to find the constants or polynomial coefficients.

5. Combine the partial fractions back together to obtain the original rational function.

For example, consider the rational function:

F(x) = (3x^2 + 2x + 1) / (x^3 - 1)

The denominator can be factored as:

x^3 - 1 = (x - 1)(x^2 + x + 1)

Thus, we can write:

F(x) = A / (x - 1) + Bx + C / (x^2 + x + 1)

where A, B, and C are constants to be determined. Equating the numerators gives:

3x^2 + 2x + 1 = A(x^2 + x + 1) + Bx(x - 1) + C(x - 1)

Solving for A, B, and C gives:

A = 1/3 B = 1/3 C = 1/3

Therefore, the partial fraction decomposition of F(x) is:

F(x) = 1/3 / (x - 1) + 1/3 x + 1/3 / (x^2 + x + 1