This problem is on partial fractions. Suppose you have
where the polynomials qi are relatively prime and all the polynomials
coefficients in a field of scalars
F. Thus there exist polynomials ai having coefficients in
Now continue doing this on each term in the above sum till finally you obtain an expression of the
Using the Euclidean algorithm for polynomials, explain why the above is of the form
where the degree of each ri is less than the degree of
M is a polynomial. Now argue
M = 0
. From this explain why the usual partial fractions expansion of calculus must be true.
You can use the fact that every polynomial having real coefficients factors into a product of
irreducible quadratic polynomials and linear polynomials having real coefficients. This follows from
the fundamental theorem of algebra.