\( - 9 x ^ { 3 } - 3 x ^ { 2 } + 3 z + 1 = \)

To solve the equation \( - 9x^{3} - 3x^{2} + 3z + 1 = 0 \), we can rearrange it to isolate \( z \). By doing so, we get \( 3z = 9x^{3} + 3x^{2} - 1 \). Then, dividing everything by 3 gives \( z = 3x^{3} + x^{2} - \frac{1}{3} \). This expresses \( z \) in terms of \( x \), making it easier to analyze or graph. Remember that when dealing with polynomial equations like this, it's often useful to graph \( z \) as a function of \( x \). Sketching or using graphing software helps visualize how changes in \( x \) influence \( z \), offering insights into local maxima, minima, or intercepts. Plus, you can spot possible roots of the polynomial that might not be immediately obvious!