18) \( 3 n^{3}-2 n^{2}-9 n+6 \)

Let's dive into the intriguing world of polynomials! The expression \(3 n^{3}-2 n^{2}-9 n+6\) is a cubic polynomial, meaning its highest degree is 3, which tells us it has the potential for three roots. Did you know that polynomials like this can be plotted on a graph, showing how it behaves as \(n\) varies? It can have turning points and cross the x-axis up to three times, depending on the discriminant! When tackling a polynomial like this, one common approach is to look for rational roots using the Rational Root Theorem. This involves checking factors of the constant term (6) over factors of the leading coefficient (3). Remember, substituting values such as \(n = 1, -1, 2, -2\), etc., could reveal roots. Also, pay attention to signs; they can help you track down mistakes or lead you towards more efficient solutions. Just make sure you check your work!