Factorise the quartic polynomial g(x)=2x4−x3−15x2−4x+12 completely over the set of real numbers.
a. (i) Show that x=3 is a zero of g(x).
b. (ii) Divide g(x) by the corresponding linear factor to obtain the cubic quotient, and then factorise the quotient further until g(x) is written as a product of irreducible factors.
c. (iii) State whether any quadratic factor obtained can be split into linear factors with rational coefficients, giving reasons for your answer.