Step 1: Determine the roots of the cubic equation. The polynomial \(P(x) = 2x^3 - 11x^2 + 17x - 6\) can be factored. By factorization or synthetic division, we find:
\(P(x) = (x - 1)(2x^2 - 9x + 6)\)
Solving \(2x^2 - 9x + 6 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with a = 2, b = −9, c = 6:
\(x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(2)(6)}}{2(2)} = \frac{9 \pm \sqrt{81 - 48}}{4} = \frac{9 \pm \sqrt{33}}{4}\)
The roots are:
\(x = 1\), \(x = \frac{9 + \sqrt{33}}{4}\), \(x = \frac{9 - \sqrt{33}}{4}\).
Step 2: Assign roots as radii and compute their squares. The areas of circles are proportional to the squares of their radii. Let the radii be:
\(r_1 = \frac{9 + \sqrt{33}}{4}\), \(r_2 = \frac{9 - \sqrt{33}}{4}\), \(r_3 = 1\).
The squares of these radii are:
\(r_1^2 = \left( \frac{9 + \sqrt{33}}{4} \right)^2 = \frac{114 + 18\sqrt{33}}{16}\)
\(r_2^2 = \left( \frac{9 - \sqrt{33}}{4} \right)^2 = \frac{114 - 18\sqrt{33}}{16}\)
\(r_3^2 = 1^2 = 1\).
Step 3: Calculate the ratio of areas. Since areas are proportional to the squares of the radii, the ratio of the areas is equivalent to the ratio of the squares of the radii.
Ratio of areas = 9 : 4 : 1.
Answer: 9:4:1