Partial Fraction Expansion: Linear Factors

Partial Fraction Expansion (PFE) renames a fraction of polynomials using smaller, simpler ‘pieces’.
The preceding section introduces PFE, reviews all needed concepts, and presents a simple example (distinct linear factors).
This current section builds on that one, giving:

Summary: The Complete ‘How-To’ of Partial Fraction Expansion

PFE Example: Repeated Linear Factors

Find the partial fraction expansion of $\displaystyle\,\frac{3x^2 + 15x + 8}{(x+1)^2(x-3)}\,$.

Note:
Here, the denominator (a cubic polynomial) is already completely factored—rejoice!
The denominator has one distinct linear factor, $\,x-3\,$, and a repeated linear factor, $\,(x+1)^2\,$.

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