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## Main Question or Discussion Point

I am reviewing the method of partial fraction decomposition, and I get to the point that I have a matrix equation that relates the coefficients of the the original numerator to the coefficients of the numerators of the partial fractions, with the each column corresponding to a certain polynomial at an offset that is the corresponding power of the term in the numerator of a particular partial fraction.

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

D( x ) = ( x - 1 )

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

g( x ) = ( x

h( x ) = ( x - 1 )

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

[ 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?

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}- 1g( 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?