Answer the following questions regarding rational functions and their algebraic manipulation.
(i) Define a proper rational function and an improper rational function in terms of the degrees of the numerator and denominator polynomials.
(ii) For each rational function given below, state whether it is proper or improper and justify your answer:
(a) x3−x+2x2+3x+1
(b) 2x3+75x4−1
(c) x3+2x−14x3−x+5
(iii) Explain why an improper rational function must first be rewritten as the sum of a polynomial and a proper rational function before it can be expressed as a sum of simpler fractions.
(iv) Describe, without performing any calculations, how the comparison of degrees of the numerator and denominator guides the first step of this rewriting process.