# Fundamental Theorem of Algebra

Pronunciation: /ˌfʌn dəˈmɛn tl ˈθi ər əm ʌv ˈæl dʒə brə/ Explain
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.

### References

- Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pp 32-34. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 8/6/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n41/mode/1up/search/fundamental. Buy the book
- Phillips, E. G.. Functions of a Complex Variable. 8th edition. pg 113. www.archive.org. Oliver and Boyd. 1961. Last Accessed 8/6/2018. http://www.archive.org/stream/functionsofacomp029605mbp#page/n126/mode/1up/search/fundamental. Buy the book
- Benjamin Fine and Gerhard Rosenberger. The Fundamental Theorem of Algebra. Springer. June 20, 1997. Buy the book

### More Information

- McAdams, David E.. Factoring Polynomials. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/f/factoringpolynomials.html.

### Cite this article as:

McAdams, David E. Fundamental Theorem of Algebra. 7/11/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/fundamentaltheoremalgebra.html.
### Revision History

7/9/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (

McAdams, David E.)
2/5/2010: Added "References". (

McAdams, David E.)
2/6/2009: Initial version. (

McAdams, David E.)