Divide - Maple Help
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Divide

division of polynomials over algebraic extension fields

 Calling Sequence evala(Divide(P, Q, 'p'))

Parameters

 P, Q - polynomials over an algebraic number or function field p - (optional) a name

Description

 • This function returns true if the polynomial Q divides P and false otherwise. The coefficients of P and Q must be algebraic functions or algebraic numbers.
 • Algebraic functions and algebraic numbers may be represented by radicals or with the RootOf notation (see type,algnum, type,algfun, type,radnum, type,radfun).
 • When Q divides P, the optional argument p is assigned the quotient P/Q.
 • The division property is meant in the domain ${K}_{x}$ where:
 x is the set of names in P and Q which do not appear inside a RootOf or a radical,
 K is a field generated over the rational numbers by the coefficients of P and Q.
 The arguments P and Q must be polynomials in x.
 • Algebraic numbers and functions occurring in the results are reduced modulo their minimal polynomial (see Normal).
 • If a or b contains functions, their arguments are normalized recursively and the functions are frozen before the computation proceeds.
 • Other objects are frozen and considered as variables.

Examples

 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{3}+x+1\right)\right)$
 ${\mathrm{\alpha }}$ (1)
 > $\mathrm{evala}\left(\mathrm{Divide}\left({x}^{3}+x+1,x-\mathrm{\alpha },'\mathrm{a1}'\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{a1}$
 ${{\mathrm{\alpha }}}^{{2}}{+}{\mathrm{\alpha }}{}{x}{+}{{x}}^{{2}}{+}{1}$ (3)
 > $\mathrm{evala}\left(\mathrm{Divide}\left({x}^{3}+x+1,x-\mathrm{\alpha }+1,'\mathrm{a2}'\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{a2}$
 ${\mathrm{a2}}$ (5)
 > $\mathrm{P1}≔\mathrm{expand}\left(\left(x-\mathrm{sqrt}\left(t\right)y\right){\left(\mathrm{sqrt}\left(t\right)x+1\right)}^{2}\right)$
 ${\mathrm{P1}}{≔}{t}{}{{x}}^{{3}}{+}{2}{}\sqrt{{t}}{}{{x}}^{{2}}{+}{x}{-}{{t}}^{{3}}{{2}}}{}{y}{}{{x}}^{{2}}{-}{2}{}{t}{}{y}{}{x}{-}\sqrt{{t}}{}{y}$ (6)
 > $\mathrm{Q1}≔ty-\mathrm{sqrt}\left(t\right)x$
 ${\mathrm{Q1}}{≔}{t}{}{y}{-}\sqrt{{t}}{}{x}$ (7)
 > $\mathrm{evala}\left(\mathrm{Divide}\left(\mathrm{P1},\mathrm{Q1},'\mathrm{a3}'\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{a3}$
 ${-}\sqrt{{t}}{}{{x}}^{{2}}{-}{2}{}{x}{-}\frac{{1}}{\sqrt{{t}}}$ (9)

The second argument below is not a polynomial. Therefore, an error is returned:

 > $\mathrm{evala}\left(\mathrm{Divide}\left(\mathrm{P1},\mathrm{Q1}+\frac{1}{x},'\mathrm{a3}'\right)\right)$