Mathematics

Problem

Let

E(x)=x^{3}+\dfrac{2}{x^{k}}-5x+7

, where

k

is an integer. a. (i) Determine all integral values of

k

for which

E(x)

is a polynomial in

x

. Give justification for each value. b. (ii) For every permissible value of

k

, write the degree of the corresponding polynomial. c. (iii) Find the value(s) of

k

for which the degree of

E(x)

becomes

5

, and write the resulting polynomial explicitly.