Question 80
Let be given two polynomials
$$
P(x) = 4x^3 - 2x^2 - 15x + 9
$$
and
$$ Q(x) = 12x^3 + 6x^2 - 7x + 1. $$
1/ Prove that each of these polynomials has three distinct real roots.
2/ Let $\alpha$ and $\beta$ be respectively the greatest roots of $P(x)$ and $Q(x)$. Prove that $\alpha^2 + 3\beta^2 = 4$.
$$
P(x) = 4x^3 - 2x^2 - 15x + 9
$$
and
$$ Q(x) = 12x^3 + 6x^2 - 7x + 1. $$
1/ Prove that each of these polynomials has three distinct real roots.
2/ Let $\alpha$ and $\beta$ be respectively the greatest roots of $P(x)$ and $Q(x)$. Prove that $\alpha^2 + 3\beta^2 = 4$.