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roots

exact roots of a polynomial with respect to one variable

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

roots(a, x, K)

Parameters

a

-

polynomial (either univariate or in x)

K

-

(optional) algebraic number field extension

x

-

(optional) polynomial variable

Description

• 

The roots function computes the exact roots of a polynomial over the rationals or an algebraic number field. The roots are returned as a list of pairs of the form [[r1,m1],...,[rn,mn]] where ri is a root of the polynomial a with multiplicity mi, that is, xrimi divides a.

• 

The call roots(a) returns roots over the field implied by the coefficients present.  For example, if all the coefficients are rational, then the rational roots are computed.  If a has no roots in the implied coefficient field, then an empty list is returned.  This assumes that a is a univariate polynomials.

• 

The call roots(a, K) computes the roots of a over the algebraic number field defined by K. Here K must be a single RootOf, or a list or set of RootOfs, or a single radical, or a list or set of radicals.  For example, if I is given as the second argument, then roots looks for the roots of a over the complex rationals.

• 

The calls roots(a, x) and roots(a, x, K) are equivalent to the above if a is univariate in x. Otherwise, it treats the other indeterminates in a as parameters, and finds all roots as above and ignoring symbolic roots.

Examples

roots2x3+11x2+12x9

3,2,12,1

(1)

rootsx44

(2)

rootsx44,x

(3)

rootsx3+6bax2+6a+5+5b+abx5a5ab,x

5,1

(4)

rootsx44,2

2,1,2,1

(5)

rootsx44,2,I

2,1,I2,1,I2,1,2,1

(6)

aliasα=RootOfx22:

aliasβ=RootOfx2+2:

rootsx44xa,x,α

α,1,α,1

(7)

rootsx44,α,β

β,1,β,1,α,1,α,1

(8)

See Also

factor

realroot

root

RootOf

Roots

solve

sturm

sturmseq

 


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