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Fundamental Theorem of Algebra

Undergraduate

Definition

The Fundamental Theorem of Algebra states every polynomial of degree n≥1 has exactly n roots (counting multiplicity) in complex numbers.

p(z) = aₙ(z-r₁)(z-r₂)·s(z-rₙ)

Factorization of degree n polynomial

Example 1

Find the roots of x² + 1 = 0.

Example 2

Find all roots of x³ - 1 = 0.

Discovered by: Carl Friedrich Gauss (1799)

Gauss gave the first rigorous proof in his doctoral dissertation.

Transfer function poles

Filter design

Polynomial root finding