91Ó°ÊÓ

Find the roots of \(x^{4}-15 x^{2}-20 x-6\). Answer: \(-3,-1,2 \pm \sqrt{6}\).

(i) Prove that \(\cosh (3 \theta)=4 \cosh ^{3}(\theta)-3 \cosh (\theta)\). (ii) Prove that \(\sinh (3 \theta)=4 \sinh ^{3}(\theta)+3 \sinh (\theta)\).

(i) Find the roots of \(x^{3}-15 x-4\) using the cubic formula. (ii) Find the roots using the trigonometric formula.

Let \(f(x) \in \mathbb{Q}[x]\) be an irreducible cubic with Galois group \(G\). (i) Prove that if \(f(x)\) has exactly one real root, then \(G \cong S_{3}\). (ii) Find the Galois group of \(f(x)=x^{3}-2 \in \mathbb{Q}[x]\). (iii) Find a cubic polynomial \(g(x) \in \mathbb{Q}[x]\) whose Galois group has order 3 .