Mathematics

Problem

Consider the family of rational functions

R_n(x)=\frac{P(x)}{\bigl(x^{2}+4x+5\bigr)^{n}},\qquad n\ge 1,

where

P(x)

is a polynomial satisfying

\deg P<2n

. Answer the following questions. a. (i) State the condition on

\deg P(x)

that guarantees

R_n(x)

is a proper rational function. b. (ii) Give a reason why, whenever

R_n(x)

is rewritten as a finite sum of terms whose denominators are the successive powers

(x^{2}+4x+5)^k

for

k=1,2,\dots ,n

, the numerator attached to each such denominator can always be chosen to have degree at most

1

. c. (iii) For

n=3

, suppose

R_3(x)

is expressed in the form

\sum_{k=1}^{3}\frac{A_kx+B_k}{(x^{2}+4x+5)^{k}}.

Determine the maximum possible number of unknown real constants that must be solved for in such a representation.