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Polynomial Factorization

A factor of a polynomial P(x) of degree n is a polynomial Q(x) of degree less than n which can be multiplied by another polynomial R(x) of degree less than n to yield P(x), i.e., a polynomial Q(x) such that

P(x)=Q(x)R(x).

For example, since

x^2-1=(x+1)(x-1),

both x-1 and x+1 are factors of x^2-1.

Polynomial factorization can be performed in the Wolfram Language using Factor[poly]. Factorization over an algebraic number field is implemented as Factor[poly, Extension -> ext].

The coefficients of factor polynomials are often required to be real numbers or integers but could, in general, be complex numbers. The fundamental theorem of algebra states that a polynomial P(z) of degree n has n values z_i (some of which are possibly degenerate) for which P(z_i)=0. Such values are called polynomial roots.

The average number of factors of a polynomial p=sum_(k=0)^(d)c_kx^k of degree d with integer coefficients c_k in the range -f<=c_k<=f has been considered by Schinzel (1976), Pinner and Vaaler (1996), Bérczes and Hajdu (1998), and Dubickas (1999).

See also

AC Method, Berlekamp-Zassenhaus Algorithm, Factor, Factorization, Fundamental Theorem of Algebra, Kronecker's Algorithm, Polynomial Factor Theorem, Polynomial Roots, Prime Factorization

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References

Abbott, J.; Shoup, V.; and Zimmermann, P. "Factorization in Z[x]: The Searching Phase." In Proceedings of the 2000 international Symposium on Symbolic and Algebraic Computation (St. Andrews, Scotland) (Ed. C. Traverso). New York: ACM, pp. 1-7, 2000.Bérczes, A. and Hajdu, L. "On a Problem of P. Turán Concerning Irreducible Polynomials." In Number Theory. Diophantine, Computational and Algebraic Aspects. Proceedings of the International Conference held in Eger, July 29-August 2, 1996. (Ed. K. Győry, A. Pethő, and V. T. Sós). Berlin: de Gruyter, pp. 95-100, 1998.Dubickas, A. "On a Polynomial with Large Number [sic] of Irreducible Factors." In Number theory in progress, Vol. 1. Diophantine Problems and Polynomials. Proceedings of the International Conference on Number Theory held in Honor of Andrzej Schinzel on his 60th Birthday in Zakopane-Kościelisko, June 30-July 9, 1997 (Ed. K. Győry, H. Iwaniec, and J. Urbanowicz). Berlin: de Gruyter, pp. 103-110, 1999.Kaltofen, E. "Polynomial Factorization." In Computer Algebra: Symbolic and Algebraic Computation, 2nd ed. (Ed. B. Buchberger, G. E.Collins, R. Loos, and R. Albrecht). Vienna: Springer-Verlag, pp. 95-113, 1983.Lenstra, A. K.; Lenstra, H. W.; and Lovász, L. "Factoring Polynomials with Rational Coefficients." Math. Ann. 261, 515-534, 1982.Pinner, C. G. and Vaaler, J. D. "The Number of Irreducible Factors of a Polynomial. II." Acta Arith. 78, 125-142, 1996.Schinzel, A. "On the Number of Irreducible Factors of a Polynomial." In Topics in Number Theory. Proceedings of the Colloquium held in Debrecen from 3-7 October, 1974. (Ed. P. Turán). Amsterdam, Netherlands: North Holland, pp. 305-314, 1976.Séroul, R. "Factoring a Polynomial with Integral Coefficients." §10.14 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 286-295, 2000.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. http://www.mathematicaguidebooks.org/.van Hoeij, M. "Factoring Polynomials and the Knapsack Problem." Preprint. http://www.math.fsu.edu/~aluffi/archive/paper124.ps.gz.

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Polynomial Factorization

Cite this as:

Weisstein, Eric W. "Polynomial Factorization." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolynomialFactorization.html

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