“Polynomial Root”的概念、定义、习题解题方法及翻译-数学辞典

单词

Polynomial Root

释义

Polynomial Root

If the Coefficients of the Polynomial

(1)

are specified to be Integers, then integral roots must have a Numerator which is a factor of

and a Denominator which is a factor of

(with either sign possible). This is known as the PolynomialRemainder Theorem.

Let the Roots of the polynomial

(2)

be denoted

, ...,

. Then Newton's Relations are

			(3)
			(4)
			(5)

These can be derived by writing

(6)

expanding, and then comparing the coefficients with (2).

Any Polynomial can be numerically factored, although different Algorithms have differentstrengths and weaknesses.

If there are no Negative Roots of a Polynomial (as can be determined by Descartes' SignRule), then the Greatest Lower Bound is 0. Otherwise, write out the Coefficients, let ,and compute the next line. Now, if any Coefficients are 0, set them to minus the sign of the nexthigher Coefficient, starting with the second highest order Coefficient. If all the signs alternate, is thegreatest lower bound. If not, then subtract 1 from , and compute another line. For example, consider the Polynomial

(7)

Performing the above Algorithm then gives

0	2	2		1
	2	0		8
--	2			8
	2			7
	2		5		35

so the greatest lower bound is .

If there are no Positive Roots of a Polynomial (as can be determined by Descartes' Sign Rule),the Least Upper Bound is 0. Otherwise, write out the Coefficients of thePolynomials, including zeros as necessary. Let . On the line below, write the highest orderCoefficient. Starting with the second-highest Coefficient, add times the number just written to the originalsecond Coefficient, and write it below the second Coefficient. Continue through order zero. If all theCoefficients are Nonnegative, the least upper bound is . If not, add one to and repeatthe process again. For example, take the Polynomial

(8)

Performing the above Algorithm gives

0	2			1
1	2	1
2	2	3
3	2	5	8	25	68

so the Least Upper Bound is 3.

See also Bairstow's Method, Descartes' Sign Rule, Jenkins-Traub Method, Laguerre's Method,Lehmer-Schur Method, Maehly's Procedure, Muller's Method, Root,Zassenhaus-Berlekamp Algorithm

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