Mathematics

Problem

Let

f(x)=x^{4}-4x^{3}+ax^{2}+bx+12

, where

a

and

b

are real constants. a. (i) Find the remainder when

f(x)

is divided by

x-3

in terms of

a

and

b

. b. (ii) If this remainder equals

5

, obtain the relation between

a

and

b

. c. (iii) Given further that

f(1)=0

, determine the values of

a

and

b