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Every real polynomial of odd degree has at least one real number as a root.
Many real polynomials of even degree do not have a real root, but the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities.
The non-real roots of polynomials with real coefficients come in conjugate pairs.
Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

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