Q10
The roots of the polynomial P(x) = 2x3 - 11x2 + 17x - 6 are the radii of three concentric circles. The ratio of their area, when arranged from the largest to the smallest, is:
Answer: D
Find the roots: try simple values.
- x = 2: 16 − 44 + 34 − 6 = 0, so (x − 2) is a factor.
- Divide: \( 2x^3 - 11x^2 + 17x - 6 = (x - 2)(2x^2 - 7x + 3) = (x - 2)(2x - 1)(x - 3) \)
- Roots (radii): 3, 2, 1/2.
Check with Vieta: sum = 11/2 = 5.5 = 3 + 2 + 0.5, and product = 6/2 = 3 = 3 × 2 × 0.5.
Areas are in the ratio of the squares of the radii:
\( 9 : 4 : \tfrac{1}{4} = 36 : 16 : 1 \)
Answer: 36 : 16 : 1 (Option 4)