Question 53
A Mediterranean polynomial has only real roots and it is of the form
$$P(x) = x^{10} - 20x^9 + 135x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0,$$ with real coefficients $a_0, a_1, \ldots, a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
$$P(x) = x^{10} - 20x^9 + 135x^8 + a_7x^7 + a_6x^6 + a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0,$$ with real coefficients $a_0, a_1, \ldots, a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.