notes/education/math/Partial Fractions.md

Partial fraction decomposition is when you break a polynomial fraction down into smaller fractions that add together.

Degree where the numerator is less

1. Factor the bottom.
2. Create two fractions, \frac{a}{p1}, and \frac{b}{p2}, where p1 and p2 are the polynomials you just factored out, and a/b are arbitrary variables
3. Multiply a by p2, and b by p1., giving you: $$\frac{ap2}{p1} + \frac{bp1}{p2}
4. When you split the

Example

 \frac{2x+1}{(x+1)(x+2)} 

Given the above fraction, the denominator is already factored, so we can move onto the next step, where two fractions are made with a and b in the numerator, and each of the denominator components in the denominator:

 \frac{a}{x+1} + \frac{b}{x+2} 

Next, find a common denominator so that you can add the two fractions together. In this case, it's (x+1)(x+2).

Degree of the numerator is equal

1. First perform polynomial division.
2. Then find a partial fraction with the remainder

Degree where the numerator is greater

1. First perform polynomial division to reach a point where the degree of the numerator is less than the degree of the denominator