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 sdiv(poly1,poly2[,v])

 sdivm(poly1,poly2,mod[,v])

:: Quotient of poly1 divided by poly2 provided that the
division can be performed within polynomial arithmetic over the
rationals.
 srem(poly1,poly2[,v])

 sremm(poly1,poly2,mod[,v])

:: Remainder of poly1 divided by poly2 provided that the
division can be performed within polynomial arithmetic over the
rationals.
 sqr(poly1,poly2[,v])

 sqrm(poly1,poly2,mod[,v])

:: Quotient and remainder of poly1 divided by poly2 provided
that the division can be performed within polynomial arithmetic over
the rationals.
 return

sdiv()
, sdivm()
, srem()
, sremm()
: polynomial sqr()
, sqrm()
: a list [quotient,remainder]
 poly1 poly2

polynomial
 v

indeterminate
 mod

prime

Regarding poly1 as an univariate polynomial in the main variable
of poly2,
i.e. var(poly2) (v if specified),
sdiv()
and
srem()
compute
the polynomial quotient and remainder of poly1 divided by poly2.
sdivm()
, sremm()
, sqrm()
execute the same
operation over GF(mod).

Division operation of polynomials is performed by the following steps:
(1) obtain the quotient of leading coefficients; let it be Q;
(2) remove the leading term of poly1 by subtracting, from
poly1, the product of Q with some powers of main variable
and poly2; obtain a new poly1;
(3) repeat the above step until the degree of poly1 become smaller
than that of poly2.
For fulfillment, by operating in polynomials, of this procedure, the
divisions at step (1) in every repetition must be an exact division of
polynomials. This is the true meaning of what we say
"division can be performed within polynomial arithmetic
over the rationals."

There are typical cases where the division is possible:
leading coefficient of poly2 is a rational number;
poly2 is a factor of poly1.

Use
sqr()
to get both the quotient and remainder at once.

Use
idiv()
, irem()
for integer quotient.

For remainder operation on all integer coefficients, use
%
.
[0] sdiv((x+y+z)^3,x^2+y+a);
x+3*y+3*z
[1] srem((x+y+z)^2,x^2+y+a);
(2*y+2*z)*x+y^2+(2*z1)*y+z^2a
[2] X=(x+y+z)*(xyz)^2;
x^3+(yz)*x^2+(y^22*z*yz^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
[3] Y=(x+y+z)^2*(xyz);
x^3+(y+z)*x^2+(y^22*z*yz^2)*xy^33*z*y^23*z^2*yz^3
[4] G=gcd(X,Y);
x^2y^22*z*yz^2
[5] sqr(X,G);
[xyz,0]
[6] sqr(Y,G);
[x+y+z,0]
[7] sdiv(y*x^3+x+1,y*x+1);
divsp: cannot happen
return to toplevel
 References

section
idiv
, irem
, section %
.
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