Question 3
Let $P(x)$ be a polynomial of degree $4$ having four positive roots. Prove that
$$\frac{1-4x}{x^2}P(x)+\left(1-\frac{1-4x}{x^2}\right)P'(x)-P''(x)=0$$
also has four positive roots. If the four given roots are distinct, the four roots obtained below are distinct; the multiplicity version is also explained.
$$\frac{1-4x}{x^2}P(x)+\left(1-\frac{1-4x}{x^2}\right)P'(x)-P''(x)=0$$
also has four positive roots. If the four given roots are distinct, the four roots obtained below are distinct; the multiplicity version is also explained.