Question

1. $6n^{3}+27n^{2}+9$

Explanation:

Step1: Find the greatest common factor (GCF)

The terms are \(6n^3\), \(27n^2\), and \(9\). The GCF of the coefficients \(6\), \(27\), and \(9\) is \(3\), and there is no common variable factor (since the last term has no \(n\)). So we factor out \(3\):
\(6n^3 + 27n^2 + 9 = 3(2n^3 + 9n^2 + 3)\)

Step2: Check if the cubic polynomial can be factored further

The cubic polynomial \(2n^3 + 9n^2 + 3\) has no rational roots (by Rational Root Theorem: possible roots are \(\pm1\), \(\pm3\), \(\pm\frac{1}{2}\), \(\pm\frac{3}{2}\), and none satisfy the equation \(2n^3 + 9n^2 + 3 = 0\)). So it doesn't factor over the rationals.

Answer:

\(3(2n^3 + 9n^2 + 3)\)