Properties
- As stated above, the GCD of two polynomials exists if the coefficients belong either to a field, the ring of the integers or more generally to a unique factorization domain.
- If c is any common divisor of p and q, then c divides their GCD.
- for any polynomial r. This property is at the basis of the proof of Euclid's algorithm.
- For any invertible element k of the ring of the coefficients, .
- Hence for any scalars such that is invertible.
- If, then .
- If, then .
- For two univariate polynomials p and q over a field, there exist polynomials a and b, such that and divides every such linear combination of p and q (Bézout's identity).
- The greatest common divisor of three or more polynomials may be defined similarly as for two polynomials. It may be computed recursively from GCD's of two polynomials by the identities:
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- and
Read more about this topic: Greatest Common Divisor Of Two Polynomials
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)