The fundamental theorem of algebra states that every non-constant single variable polynomial with complex coefficients has at least one complex root.^{[1]} More importantly, for the level of math covered by this encyclopedia, the fundamental theorem of algebra implies that:
Every non-constant polynomial with real coefficients can be factored over the real numbers into a product of linear factors and irreducible quadratic factors.A linear factor is a factor in the form bx+c where b and c are real numbers. A quadratic factor is a factor in the form ax^{2}+bx+c where a, b and c are real numbers. A quadratic factor is irreducible if it can not be factored into two linear factors. The discriminant of a quadratic factor tells if the quadratic factor is reducible. If the discriminant is less than zero, the quadratic factor is irreducible. If the discriminant is greater or equal to zero, the quadratic factor can be reduce to two linear factors.
The fundamental theorem of algebra is an existence theorem. While the fundamental theorem of algebra proves that the factors exist, it does not tell how to find the factors. For more information on factoring polynomials, see Factoring Polynomials.
When the fundamental theorem of algebra was named, algebra consisted mainly of the study of polynomials. The fundamental theorem of algebra is not fundamental to modern algebra.
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