Question 5
Define the polynomials
$$p(x)=4x^3-2x^2-15x+9,\qquad q(x)=12x^3+6x^2-7x+1.$$
Prove that each polynomial has at least three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Prove that $A^2+3B^2=4$.
$$p(x)=4x^3-2x^2-15x+9,\qquad q(x)=12x^3+6x^2-7x+1.$$
Prove that each polynomial has at least three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Prove that $A^2+3B^2=4$.