Please explain how Partial Fraction Decomposition and how to solve for the constants for linear, repeated linear, and irreducible quadratic factors.
Ok, then. Um, how do I turn a partial fraction decomposition into a system of equations to use as an augmented matrix to find the corresponding constants?
Example problem:
[ -(x^2) + 2x + 4 ] / [ x^3 - 4x^2 + 4x ]
=
A1/x + A2/(x-2) + A3/[ (x-2)^2 ]
These are the decomposition factors, but what do I do now to find the constants?
$\displaystyle \frac {-x^2 + 2x + 4}{x^3 - 4x^2 + 4x} = \frac Ax + \frac B{x - 2} + \frac C{(x - 2)^2}$
multiply both sides by the denominator of the left hand side:
$\displaystyle \Rightarrow -x^2 + 2x + 4 = A(x - 2)^2 + Bx(x - 2) + Cx$
let $\displaystyle x = 0:~\Rightarrow 4 = 4A$
let $\displaystyle x = 2:~ \Rightarrow 4 = 2C$
let $\displaystyle x = 1:~\Rightarrow 5 = A - B + C$
there you go
look at the expression we have. note that if x = 0 the terms multiplying B and C become zero, and hence i am just left with A to solve for. similarly, if x = 2, A and B go away, and i am left with C. x = 1 was a random number i used to get a third equation. i just picked something small enough to make computation easy. i would have already found two unknowns from the first two equations, so this last one will help me find the third