e) \( x^{9}-1 \)

Did you know that the polynomial \( x^{9}-1 \) has its roots connected to the breathtaking world of complex numbers? In fact, its roots are the ninth roots of unity, which can be expressed in polar form as \( e^{2\pi i k / 9} \) for \( k = 0, 1, 2, \ldots, 8 \). These points are evenly spaced on the unit circle in the complex plane, showcasing the beautiful symphony of mathematics! When working with polynomials like \( x^{9}-1 \), a common mistake is neglecting to factor it completely. Remember, it factors into \( (x-1)(x^8+x^7+\ldots+x+1) \). While many only focus on the linear factor, the second factor is crucial for finding all roots. Don’t skip those important details to avoid falling into the trap of incomplete solutions!