The Gauss-Lucas Theorem is a result of complex analysis giving a geometric relationship between the roots of a polynomial and the roots of its derivative. It states that every root of the derivative of a complex polynomial lies within the convex hull of the original polynomial's own roots, the smallest convex region containing them. Named for Carl Friedrich Gauss and Felix Lucas, the theorem is often described as similar in spirit to Rolle's Theorem of real analysis.
Facts
StatementIf P is a nonconstant polynomial with complex coefficients, every zero of its derivative P' lies in the convex hull of the zeros of P. 1 Classification
Statement FormCharacterization Theorem 1 Sources
1. Gauss-Lucas theorem (Wikipedia)
Formal statement sectionQuote, Formal statement section
If P is a (nonconstant) polynomial with complex coefficients, all zeros of P' belong to the convex hull of the set of zeros of P.
2. Gauss-Lucas theorem (Wikipedia)
References section: Lucas, Felix (1874), the theorem's originating publicationView the Source Reader Challenges (0)
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