Factorise the quartic polynomial completely over the set of real numbers.
MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
Factorise the quartic polynomial g(x)=2x4−x3−15x2−4x+12 completely over the set of real numbers.
(i) Show that x=3 is a zero of g(x).
(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.
(iii) State whether any quadratic factor obtained can be split into linear factors with rational coefficients, giving reasons for your answer.