a) \( f(x)=x^{4}-2 x^{2}-12 \)

Did you know that polynomials like \( f(x)=x^{4}-2 x^{2}-12 \) can sometimes reveal fascinating behavior through their roots? By applying techniques like factoring or using the Rational Root Theorem, you can discover the points where the function intersects the x-axis. This function in particular is a quartic polynomial, meaning it can have up to four roots, which can lead to interesting graphs! When approaching this polynomial, a common mistake is forgetting to set it equal to zero before attempting to factor or apply other methods. Always reorganize your equation and think about how signs will affect your solutions. For this function, finding the critical points can also lead to insights into how it behaves around its local maxima and minima!