If the option 'symbolic'=true is given and a RootOf whose minimal polynomial factors nontrivially is detected, then Divide will reduce it to a RootOf of lower degree by picking one of the factors arbitrarily. This will eliminate the possibility of a "reducible RootOf detected" error. The default is 'symbolic'=false.

If the option 'characteristic'=p is given, where p is a nonnegative integer, the divisibility is computed over an extension of the ring ${\mathrm{\mathbb{Z}}}_p$ of integers modulo p. The default is 'characteristic'=0 and means that the divisibility is computed over an extension of the rational numbers.

Note that if p is positive but not a prime, then ${\mathrm{\mathbb{Z}}}_p$ is not a field, so Divide may not be able to determine whether a is divisible by b or not. If it does not succeed because it encounters an integer that has no inverse modulo p, it issues the error "zero divisor modulo p detected"

If the option 'makeindependent'=true is given, then Divide will always try to find a field representation for algebraic numbers in the input, regardless of how many algebraic objects the input contains. If the input contains many RootOfs, then this can be a very expensive calculation. If 'makeindependent'=false is given, then no independence checking is performed. The default is 'makeindependent'=FAIL, in which case algebraic dependencies will only be checked for if there are $4$ or fewer algebraic objects in the input.

The Divide command determines if a given polynomial a is exactly divisible by the polynomial b.  Formally, a is divisible by b if there exists a polynomial q such that $a=qb$.  a and b may be polynomials in one or more variables.

The inputs a and b may contain algebraic number coefficients. These may be represented by radicals or with the RootOf notation (see type,algnum, type,radnum). In general, algebraic numbers will be returned in the same representation as they were received in. Nested radicals and RootOfs are also supported.

The division property must hold in the domain K[x], where:

x is the set of names in a and b which do not appear inside a RootOf or a radical.

K is an algebraic field generated over the rationals and any algebraic number coefficients occurring in a and b (unless the option 'characteristic' is given; see below).

Note that this function does not determine divisibility over the integers, as by default, all division is computed over the field of rationals, so true will be returned if two non-zero constants are given as input. For integer division, use iquo and irem.

If the optional argument q is included and a is divisible by b, then q will be assigned the value of the quotient $\frac{a}{b}$.  It is always guaranteed that $a=qb$. If a is not divisible by b, then q will not be assigned any value.

Non-algebraic sub-expressions such as $\sin \left(x\right)$ that are neither variables, rational numbers, or algebraic objects are frozen and temporarily replaced by new local variables, which are not considered to be constant in what follows.

The inputs a and b can even be polynomials disguised as rational functions, in which case they are normalized first using Algebraic[Normal].  

If the quotient q is assigned a value, it will be normalized as follows:

All non-constant factors in q are monic with respect to a block-lexicographic ordering of the variables, where all global variables are considered larger than all local ones. If all variables have different names, then this ordering is session-independent.

There are at most two constant factors, and at most one of them is not a rational number.

All factors that are not rational numbers have integer content equal to $1$, except possibly if there is only one non-constant factor.

All algebraic numbers occurring in the result are reduced modulo their minimal polynomial (see Reduce), and all arguments of functions, if any, are normalized recursively (see Normal).

If the set of radicals and RootOfs in the input cannot be embedded into a field algebraically, then Divide may not be able to determine divisibility.  Divide will try to find a field representation if there are at most $4$ algebraic objects in the input (unless option 'makeindependent' is given; see below), and otherwise attempt to proceed anyway. If unsuccessful, a "reducible RootOf detected" error will be returned. (unless the option 'symbolic'=true is given; see below).

This function does not support input containing floats or radical functions such as $\sqrt{x}$.

$\mathrm{with}\left(\mathrm{Algebraic}\right)\:$

$\mathrm{Divide}\left(x^2+1\,x+\mathrm{I}\,'\mathrm{q1}'\right)$

$\mathrm{true}$

$\mathrm{q1}$

$x-\mathrm{I}$

$\left[\mathrm{Divide}\left(x^2+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}-1\right)x+1+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}-1\right)\,x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}-1\right)\,'\mathrm{q2}'\right)\,\mathrm{q2}\right]$

$\left[\mathrm{true}\,x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}-1\right)\right]$

$\left[\mathrm{Divide}\left(x^2+3y^2\,x+\mathrm{I}\mathrm{sqrt}\left(3\right)y\,'\mathrm{q3}'\right)\,\mathrm{q3}\right]$

$\left[\mathrm{true}\,-\mathrm{I}\sqrt{3}y+x\right]$

$\left[\mathrm{Divide}\left(x^2+y^2\,x+y\,'q'\right)\,q\right]$

$\left[\mathrm{false}\,q\right]$

$\left[\mathrm{Divide}\left(x^2+\mathrm{sqrt}\left(2\right)x+\mathrm{sqrt}\left(3\right)x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-6\,\mathrm{index}=1\right)\,x+\mathrm{sqrt}\left(2\right)\,'\mathrm{q4}'\right)\,\mathrm{q4}\right]$

$\left[\mathrm{true}\,x+\sqrt{3}\right]$

$\left[\mathrm{Divide}\left(x^2+\mathrm{sqrt}\left(2\right)x+\mathrm{sqrt}\left(3\right)x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-6\right)\,x+\mathrm{sqrt}\left(2\right)\,'\mathrm{q5}'\right)\,\mathrm{q5}\right]$

$\left[\mathrm{true}\,x+\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2-6\right)\sqrt{2}}{2}\right]$

$\mathrm{Divide}\left(x^2+\mathrm{sqrt}\left(2\right)x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right)x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-6\right)\,x+\mathrm{sqrt}\left(2\right)\right)$

$\mathrm{false}$

$\left[\mathrm{Divide}\left(x^2-4^{\frac{1}{3}}\,x+\mathrm{RootOf}\left({\mathrm{\_Z}}^3-2\,\mathrm{index}=1\right)\,'\mathrm{q6}'\right)\,\mathrm{q6}\right]$

$\left[\mathrm{true}\,x-2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]$

$\left[\mathrm{Divide}\left(xy\mathrm{sqrt}\left(\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\,\mathrm{index}=1\right)\right)\,y\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\,\mathrm{index}=1\right)\,\mathrm{index}=1\right)\,'\mathrm{q7}'\right)\,\mathrm{q7}\right]$

$\left[\mathrm{true}\,x\right]$

$\left[\mathrm{Divide}\left(17\,5\,'\mathrm{q8}'\right)\,\mathrm{q8}\right]$

$\left[\mathrm{true}\,\frac{17}{5}\right]$

$\left[\mathrm{iquo}\left(17\,5\right)\,\mathrm{irem}\left(17\,5\right)\right]$

$\left[3\,2\right]$

$\left[\mathrm{Divide}\left(x^2+y+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+5\right)\,4\,'\mathrm{q9}'\right)\,\mathrm{q9}\right]$

$\left[\mathrm{true}\,\frac{x^2}{4}+\frac{y}{4}+\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+5\right)}{4}\right]$

$\mathrm{Divide}\left(x^2+y+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+5\right)\,0\right)$

Error, (in Algebraic:-Divide) numeric exception: division by zero

$\left[\mathrm{Divide}\left(\tan \left(4\right)\mathrm{sqrt}\left(2\right)+\tan \left(4\right)x\,\tan \left(4\right)\,'\mathrm{q10}'\right)\,\mathrm{q10}\right]$

$\left[\mathrm{true}\,x+\sqrt{2}\right]$

$\left[\mathrm{Divide}\left({\sin \left(x\right)}^2-3{\mathrm{foo}\left(y\right)}^4\,\sin \left(x\right)-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right){\mathrm{foo}\left(y\right)}^2\,'\mathrm{q11}'\right)\,\mathrm{q11}\right]$

$\left[\mathrm{true}\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right){\mathrm{foo}\left(y\right)}^2+\sin \left(x\right)\right]$

$\left[\mathrm{Divide}\left(\frac{x}{2}+\sin \left(\frac{\mathrm{\pi }}{4}\right)y\,x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\,\mathrm{index}=1\right)y\,'\mathrm{q12}'\right)\,\mathrm{q12}\right]$

$\left[\mathrm{true}\,\frac{1}{2}\right]$

$\left[\mathrm{Divide}\left(x^2+\exp \left(\mathrm{I}\mathrm{\pi }\right)\,x-\sec \left(\mathrm{\pi }\right)\,'\mathrm{q13}'\right)\,\mathrm{q13}\right]$

$\left[\mathrm{true}\,x-1\right]$

$\mathrm{zero}≔0$

$\mathrm{zero}≔0$

$\mathrm{Divide}\left(3\sin \left(\mathrm{zero}\right)\,\sin \left(\mathrm{zero}\right)\right)$

Error, (in Algebraic:-Divide) numeric exception: division by zero

$\left[\mathrm{Divide}\left(x^2+\sin \left(\frac{\mathrm{\pi }\left(x+\mathrm{sqrt}\left(5\right)\right)}{x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-5\,\mathrm{index}=1\right)}\right)\,x\,'\mathrm{q14}'\right)\,\mathrm{q14}\right]$

$\left[\mathrm{true}\,x\right]$

$\mathrm{Divide}\left(\frac{1}{x}\,\frac{1}{x^2}\right)$

Error, (in Algebraic:-Divide) polynom(s) with radalgnum coefficients expected

$\mathrm{Divide}\left(\frac{x^3+1}{x-1}\,x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}+1\right)\right)$

Error, (in Algebraic:-Divide) polynom(s) with radalgnum coefficients expected

$\left[\mathrm{Divide}\left(\frac{x^3+1}{x+1}\,x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}+1\right)\,'\mathrm{q15}'\right)\,\mathrm{q15}\right]$

$\left[\mathrm{true}\,x+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}+1\right)-1\right]$

$\left[\mathrm{Divide}\left(xy{\mathrm{RootOf}\left({\mathrm{\_Z}}^3-\mathrm{\_Z}+3\right)}^5\,\mathrm{sqrt}\left(2\right)y{\mathrm{RootOf}\left({\mathrm{\_Z}}^3-\mathrm{\_Z}+3\right)}^2\,'\mathrm{q16}'\right)\,\mathrm{q16}\right]$

$\left[\mathrm{true}\,\frac{x\sqrt{2}\mathrm{RootOf}\left({\mathrm{\_Z}}^3-\mathrm{\_Z}+3\right)}{2}-\frac{3\sqrt{2}x}{2}\right]$

$\mathrm{Divide}\left(x^2+2x\mathrm{sqrt}\left(y\right)+y\,x+\mathrm{sqrt}\left(y\right)\right)$

Error, (in Algebraic:-Divide) polynom(s) with radalgnum coefficients expected

$\mathrm{Divide}\left(x^2-x+0.25\,x-\frac{1}{2}\right)$

Error, (in Algebraic:-Divide) polynom(s) with radalgnum coefficients expected

$\left[\mathrm{Divide}\left(x^3-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-1\right)\,x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-1\right)\,'\mathrm{q21}'\right)\,\mathrm{q21}\right]$

$\left[\mathrm{true}\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-1\right)x+x^2+1\right]$

$\mathrm{Divide}\left(x-\mathrm{sqrt}\left(2\right)\,x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\,\mathrm{index}=1\right)\right)$

$\mathrm{true}$

$\mathrm{Divide}\left(x-\mathrm{sqrt}\left(2\right)\,x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\,\mathrm{index}=2\right)\right)$

$\mathrm{false}$

$\mathrm{Divide}\left(x-\mathrm{sqrt}\left(2\right)\,x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\right)\right)$

$\mathrm{false}$

$\mathrm{Divide}\left(x^{2}y\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}\,\mathrm{index}=1\right)x\right)$

Error, (in Algebraic:-Divide) numeric exception: division by zero

$\mathrm{Divide}\left(x^{2}y\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}\,\mathrm{index}=2\right)x\right)$

$\mathrm{true}$

$\mathrm{Divide}\left(x^{2}y\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}\right)x\right)$

Error, (in Algebraic:-Divide) reducible RootOf detected. Substitutions are {RootOf(_Z^2-_Z) = 0, RootOf(_Z^2-_Z) = 1}

$\mathrm{Divide}\left(x^{2}y\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2-\mathrm{\_Z}\right)x\,'\mathrm{symbolic}'=\mathrm{true}\right)$

$\mathrm{true}$

$\mathrm{Divide}\left(x^2+4\,x+1\right)$

$\mathrm{false}$

$\left[\mathrm{Divide}\left(x^2+4\,x+1\,'\mathrm{q22}'\,'\mathrm{characteristic}'=5\right)\,\mathrm{q22}\right]$

$\left[\mathrm{true}\,x+4\right]$

$\left[\mathrm{Divide}\left(x^2-5\,x-\mathrm{I}\mathrm{sqrt}\left(2\right)\,'\mathrm{q23}'\,'\mathrm{characteristic}'=7\right)\,\mathrm{q23}\right]$

$\left[\mathrm{true}\,\mathrm{I}\sqrt{2}+x\right]$

$\left[\mathrm{Divide}\left(2x^{2}y+8xy\,2x+8\,'\mathrm{q24}'\,'\mathrm{characteristic}'=9\right)\,\mathrm{q24}\right]$

$\left[\mathrm{true}\,xy\right]$

$\mathrm{Divide}\left(2x^{2}y+8xy\,2x+8\,'\mathrm{characteristic}'=10\right)$

Error, (in Algebraic:-Divide) zero divisor modulo 10 detected

$\mathrm{CubeRootOf}\left[-4\right]≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3+4\,\mathrm{index}=1\right)$

${\mathrm{CubeRootOf}}_{\mathrm{-4}}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3+4\,\mathrm{index}=1\right)$

$\mathrm{CubeRootOf}\left[-2\right]≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3+2\,\mathrm{index}=1\right)$

${\mathrm{CubeRootOf}}_{\mathrm{-2}}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3+2\,\mathrm{index}=1\right)$

$\mathrm{CubeRootOf}\left[2\right]≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-2\,\mathrm{index}=1\right)$

${\mathrm{CubeRootOf}}_2≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-2\,\mathrm{index}=1\right)$

$\mathrm{CubeRootOf}\left[3\right]≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-3\,\mathrm{index}=1\right)$

${\mathrm{CubeRootOf}}_3≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-3\,\mathrm{index}=1\right)$

$\mathrm{CubeRootOf}\left[4\right]≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-4\,\mathrm{index}=1\right)$

${\mathrm{CubeRootOf}}_4≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-4\,\mathrm{index}=1\right)$

$\mathrm{CubeRootOf}\left[6\right]≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-6\,\mathrm{index}=1\right)$

${\mathrm{CubeRootOf}}_6≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3-6\,\mathrm{index}=1\right)$

$\mathrm{Divide}\left(x+\mathrm{CubeRootOf}\left[4\right]\mathrm{CubeRootOf}\left[-2\right]\mathrm{CubeRootOf}\left[3\right]\,x+\mathrm{CubeRootOf}\left[-4\right]\mathrm{CubeRootOf}\left[6\right]\right)$

$\mathrm{false}$

$\left[\mathrm{Divide}\left(x+\mathrm{CubeRootOf}\left[4\right]\mathrm{CubeRootOf}\left[-2\right]\mathrm{CubeRootOf}\left[3\right]\,x+\mathrm{CubeRootOf}\left[-4\right]\mathrm{CubeRootOf}\left[6\right]\,'\mathrm{q25}'\,'\mathrm{makeindependent}'=\mathrm{true}\right)\,\mathrm{q25}\right]$

$\left[\mathrm{true}\,1\right]$

$\left[\mathrm{Divide}\left(x+\mathrm{CubeRootOf}\left[2\right]\mathrm{CubeRootOf}\left[3\right]\,x+\mathrm{CubeRootOf}\left[6\right]\,'\mathrm{q26}'\right)\,\mathrm{q26}\right]$

$\left[\mathrm{true}\,1\right]$

$\mathrm{Divide}\left(x+\mathrm{CubeRootOf}\left[2\right]\mathrm{CubeRootOf}\left[3\right]\,x+\mathrm{CubeRootOf}\left[6\right]\,'\mathrm{makeindependent}'=\mathrm{false}\right)$

$\mathrm{false}$