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:
TYPE OF FACTOR(S) IN $\,D(x)\,$ | CORRESPONDING TERM(S) IN THE PFE |
---|---|
distinct linear factor: $\,ax + b\,$, where $\,a\ne 0\,$ |
a single term and a single unknown constant: $\displaystyle\frac{\color{red}{A}}{ax+b}$ |
repeated linear factor: $\,(ax + b)^n\,$, where $\,a\ne 0\,$ and $\,n = 2,3,4,\ldots\,$ |
$\,n\,$ terms and $\,n\,$ unknown constants: $\displaystyle\frac{\color{red}{A_1}}{ax+b} + \frac{\color{red}{A_2}}{(ax+b)^2} + \cdots + \frac{\color{red}{A_n}}{(ax+b)^n}$ |
distinct irreducible quadratic factor: $\,ax^2 + bx + c\,$, where $\,a\ne 0\,$ and $\,b^2 - 4ac < 0\,$ |
a single term and $\,2\,$ unknown constants: $\displaystyle\frac{\color{red}{A}x + \color{red}{B}}{ax^2+bx + c}$ |
repeated irreducible quadratic factor: $\,(ax^2 + bx + c)^n\,$, where $\,a\ne 0\,$, $\,b^2 - 4ac < 0\,$, and $\,n = 2,3,4,\ldots\,$ |
$\,n\,$ terms and $\,2n\,$ unknown constants: $\displaystyle\frac{\color{red}{A_1}x + \color{red}{B_1}}{ax^2 +bx + c} + \frac{\color{red}{A_2}x + \color{red}{B_2}}{(ax^2+bx + c)^2} + \cdots + \frac{\color{red}{A_n}x + \color{red}{B_n}}{(ax^2 +bx + c)^n}$ |
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, $\,(x+1)^2\,$.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
IN PROGRESS |