Polynomial Roots and Concentric Circles Area Ratio

Let's find the roots of $P(x) = 2x^3 - 11x^2 + 17x - 6 = 0$. By inspection, $P(2) = 2(8) - 11(4) + 17(2) - 6 = 16 - 44 + 34 - 6 = 0$. Dividing $P(x)$ by $(x-2)$ gives $2x^2 - 7x + 3 = 0$. $(2x - 1)(x - 3) = 0 \implies x = 1/2, 3$. So the three roots (radii) are $r_1 = 3, r_2 = 2, r_3 = 1/2$. The areas of the circles are proportional to $r^2$. Ratio of areas $= 3^2 : 2^2 : (1/2)^2 = 9 : 4 : 1/4 = 36 : 16 : 1$.