- #1

swampwiz

- 455

- 37

Anyway, the certain polynomial is the product of all unique, non-unity factors - i.e., that have a Greatest Common Factor (GCF) with respect to the others as 1 - with varying levels of multiplicity; the offset can be folded into the certain polynomial by simply factoring in the unity function to get an enhanced certain polynomial (for the case of no offset, the enhanced is the same as the regular).

E.G.

original fraction: [ R( x ) / D( x ) ]

R( x ) = r

_{0}+ r

_{1}x + r

_{2}x

^{2}+ r

_{3}x

^{3}

D( x ) = ( x - 1 )

^{2}( x

^{2}+ x + 1 )

f( x ) = ( x - 1 ) ( x

^{2}+ x + 1 ) = x

^{3}- 1

g( x ) = ( x

^{2}+ x + 1 )

h( x ) = ( x - 1 )

^{2}= x

^{2}+ 2 x + 1

[ R( x ) / D( x ) ] = [ A / ( x - 1 ) ] + [ B / ( x - 1 )

^{2}] + [ ( C x + D ) / ( x

^{2}+ x + 1 ) ]

R( x ) = A f( x ) + B g( x ) + ( C + d X ) h( x )

[ M ] = [ { f( x ) } { g( x ) } { h( x ) } { x h( x ) } ] ← not actual functions

{ r } = [ M ] { A B C D }

^{T}

[ M ] :

[ 1 1 1 0 ]

[ 0 1 2 1 ]

[ 0 1 1 2 ]

[ 1 0 0 1 ]

Now, it seems that because each one of these enhanced certain polynomials are a unique product of factors that all have a GCF with respect to each other of 1 (i.e., the GCF of the factors), the coefficients of the enhanced certain polynomial cannot have any linear dependency on any other, and thus the matrix of these columns of enhanced certain polynomials is invertible (i.e., determinant not 0). However, it seems that there should somehow be some type of theorem/lemma that proves this; what is this theorem/lemma?