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**Partial Fraction decomposition** is the opposite of combining fractions with different denominators. When we decompose fractions, we "chop" the full rational function up into the individual addends.

To decompose a fraction, you must break up the denominator of the original rational into all possible factors, which will become the denominators of the addends. If a factor has a multiplicity of two or more, the original factor must be included, as well any other powered factors. An example of this is below:

\begin{align*}\frac{6x-1}{x^2 (x-1)(x^2+2)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-1}+\frac{Dx+E}{x^2+2}\end{align*}

In the numerator, there should be a "fill-in-the-blank" expression with a degree that is exactly one less than the degree of the denominator. In most cases, there will be a constant or a line (an expression in the form Ax+B) in the numerator.

After setting up the equivalent expression, combine the expression into one fraction so that the denominator of both sides are equal, notice how the variable coefficients and constants match up with numerical values from the original expression. To solve for the variables, set up systems of equations for the coefficients and constants and solve.

Example problems and practice can be found here.