To determine the IUPAC name:
1. Identify the longest continuous carbon chain, which has 8 carbons (octane).
2. Number the carbon chain to give the substituents the lowest possible locants: - Methyl groups are at positions 2, 5, and 6.
3. Arrange the substituents in alphabetical order and include their positions.
The compound is named : 2,5,6-Trimethyloctane
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: