So $ p(t) $ has four roots and is degree ≤ 3 → $ p(t) \equiv 0 $. Hence $ d(t) = t^3 $ is the only possibility.

["Understanding Why $ p(t) $, a Degree ≤ 3 Polynomial with Four Roots, Must Be Identically Zero", "In polynomial mathematics, understanding the relationship between a polynomial’s degree and the number of roots it possesses is crucial. A fundamental principle is that a non-zero polynomial of degree $ d $ can have at most $ d $ roots (counted with multiplicity). Today, we explore a compelling case: what happens when a polynomial has four roots but its degree is at most three?", "---", "### The Core Principle: Degree and Root Limitation", "Let $ p(t) $ be a polynomial with degree $ \deg(p) \leq 3 $. By the degree-root relationship, this means:", "$$\n\ ext{Number of roots } \leq \deg(p) \leq 3\n$$", "Thus, $ p(t) $ cannot have four distinct roots unless it is the zero polynomial—that is, $ p(t) \equiv 0 $.", "---", "### Why Does This Matter?", "Suppose we consider a nonzero polynomial $ p(t) $ of degree ≤ 3. Then it can have at most three roots (including multiplicities). But if $ p(t) $ has four distinct roots, this directly contradicts the degree-roots inequality. The only consistent resolution is:", "$$\np(t) \equiv 0\n$$", "---", "### The Implication for $ d(t) $ and Polynomial Derivatives", "Suppose we define a related function $ d(t) = t^3 $, which is indeed a degree-3 polynomial. It has three distinct roots (each at $ t = 0 $, counted thrice). However, if we correctly interpret the original statement—where $ p(t) $ has four actual roots (counting multiplicity)—then $ p(t) $ must vanish identically.", "But what about $ d(t) $? The expression $ d(t) = t^3 $ alone has only three roots. However, in many contexts, $ d(t) $ may refer to a polynomial that captures behavior related to $ p(t) $, such as its derivative or a normalized version.", "Crucially, if $ p(t) \equiv 0 $, then its derivative $ d(t) = p'(t) $ is also identically zero—consistent with a triple root at zero (with multiplicity).", "---", "### Conclusion", "The key insight is clear: a nonzero polynomial of degree at most 3 cannot have four roots. Therefore, if we assert that $ p(t) $ has four roots and $ \deg(p) \leq 3 $, the only resolution is $ p(t) \equiv 0 $. Under this condition, $ d(t) = t^3 $ becomes relevant only as a model of a low-degree approximation or derivative behavior—firmly anchored in the rule that zero degree allows zero polynomials.", "Thus, the only consistent polynomial satisfying $ \deg(p) \leq 3 $ with four roots (counted properly) is the zero polynomial. As a consequence, $ d(t) = t^3 $ must vanish identically in this scenario—emphasizing the strict link between degree, roots, and polynomial identity.", "---", "TL;DR: A degree ≤ 3 polynomial cannot have four roots. Hence $ p(t) \equiv 0 $, and any related polynomial like $ d(t) = t^3 $ must reflect this degeneracy, often appearing as the formal cube of a near-singular root configuration.", "---", "By honoring this foundational rule, we deepen both theoretical understanding and practical problem-solving in polynomial analysis and calculus."]