c) \( x^{32}-1 \)

The polynomial \( x^{32}-1 \) represents a rich tapestry of mathematics rooted in the concept of roots of unity. The 32nd roots of unity are the complex numbers on the unit circle in the complex plane, each spaced evenly at \( \frac{2\pi k}{32} \) for \( k = 0, 1, 2, \ldots, 31 \). This polynomial can be factored into linear factors, revealing connections to group theory and cyclic groups, encapsulating the beautiful interplay between algebra and geometry. When dealing with \( x^{32}-1 \), a common mistake is not recognizing the significance of the distinct factors it can produce. Each factor \( x - \zeta \), where \( \zeta \) is a 32nd root of unity, can lead to confusion if one attempts to group them without considering the polynomial unity context. Additionally, be cautious about forgetting to include all the roots; for \( x^{32}-1 \), there are indeed 32 unique roots!