1. ## Partial Fraction Decomposition

Please explain how Partial Fraction Decomposition and how to solve for the constants for linear, repeated linear, and irreducible quadratic factors.

2. Originally Posted by iEricKim
Please explain how Partial Fraction Decomposition and how to solve for the constants for linear, repeated linear, and irreducible quadratic factors.
this is a big question. too big for us to answer for you here. the first 5 or so links here should help. read up yourself and come back with specific questions. we offer math help here, not lessons

3. 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?

4. Originally Posted by iEricKim
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?
$\frac {-x^2 + 2x + 4}{x^3 - 4x^2 + 4x} = \frac Ax + \frac B{x - 2} + \frac C{(x - 2)^2}$

multiply both sides by the denominator of the left hand side:

$\Rightarrow -x^2 + 2x + 4 = A(x - 2)^2 + Bx(x - 2) + Cx$

let $x = 0:~\Rightarrow 4 = 4A$

let $x = 2:~ \Rightarrow 4 = 2C$

let $x = 1:~\Rightarrow 5 = A - B + C$

there you go

5. Where did you get the numbers 0, 2, and 1?

6. Originally Posted by iEricKim
Where did you get the numbers 0, 2, and 1?
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

7. Ah, ok! Thank you very much!