polynomialReduce

Reduce polynomials by division

Description

example

r = polynomialReduce(p,d) returns the Polynomial Reduction of p by d with respect to all variables in p determined by symvar. The input d can be a vector of polynomials.

example

r = polynomialReduce(p,d,vars) uses the polynomial variables in vars.

example

r = polynomialReduce(___,'MonomialOrder',MonomialOrder) also uses the specified monomial order in addition to the input arguments in previous syntaxes. Options are 'degreeInverseLexicographic', 'degreeLexicographic', or 'lexicographic'. By default, polynomialReduce uses 'degreeInverseLexicographic'.

example

[r,q] = polynomialReduce(___) also returns the quotient in q.

Examples

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Find the quotient and remainder when x^3 - x*y^2 + 1 is divided by x + y.

syms x y
p = x^3 - x*y^2 + 1;
d = x + y;
[r,q] = polynomialReduce(p,d)
r =
1
q =
x^2 - y*x

Reconstruct the original polynomial from the quotient and remainder. Check that the reconstructed polynomial equals p by using isAlways.

pOrig = expand(sum(q.*d) + r);
isAlways(p == pOrig)
ans =
logical
1

Specify the polynomial variables as the second argument of polynomialReduce.

Divide exp(a)*x^2 + 2*x*y + 1 by x - y with the polynomial variables [x y], treating a as a symbolic parameter.

syms a x y
p = exp(a)*x^2 + 2*x*y + 1;
d = x - y;
vars = [x y];
r = polynomialReduce(p,d,vars)
r =
(exp(a) + 2)*y^2 + 1

Reduce x^5 - x*y^6 - x*y by x^2 + y and x^2 - y^3.

syms x y
p = x^5 - x*y^6 - x*y;
d = [x^2 + y, x^2 - y^3];
[r,q] = polynomialReduce(p,d)
r =
-x*y
q =
[ x^3 - x*y^3, x*y^3 - x*y]

Reconstruct the original polynomial from the quotient and remainder. Check that the reconstructed polynomial equals p by using isAlways.

pOrig = expand(q*d.' + r);
isAlways(p == pOrig)
ans =
logical
1

By default, polynomialReduce orders the terms in the polynomials with the term order degreeInverseLexicographic. Change the term order to lexicographic or degreeLexicographic by using the 'MonomialOrder' name-value pair argument.

Divide two polynomials by using the lexicographic term order.

syms x y
p = x^2 + y^3 + 1;
d = x - y^2;
r = polynomialReduce(p,d,'MonomialOrder','lexicographic')
r =
y^4 + y^3 + 1

Divide the same polynomials by using the degreeLexicographic term order.

r = polynomialReduce(p,d,'MonomialOrder','degreeLexicographic')
r =
x^2 + y*x + 1

Input Arguments

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Polynomial to divide, specified as a symbolic expression or function.

Polynomials to divide by, specified as a symbolic expression or function or a vector of symbolic expressions or functions.

Polynomial variables, specified as a vector of symbolic variables.

Monomial order of divisors, specified as 'degreeInverseLexicographic', 'degreeLexicographic', or 'lexicographic'. If you specify vars, then polynomialReduce sorts variables based on the order of variables in vars.

• lexicographic sorts the terms of a polynomial using lexicographic ordering.

• degreeLexicographic sorts the terms of a polynomial according to the total degree of each term. If terms have equal total degrees, polynomialReduce sorts the terms using lexicographic ordering.

• degreeInverseLexicographic sorts the terms of a polynomial according to the total degree of each term. If terms have equal total degrees, polynomialReduce sorts the terms using inverse lexicographic ordering.

Output Arguments

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Remainder of polynomial division, returned as a symbolic polynomial.

Quotient of polynomial division, returned as a symbolic polynomial or a vector of symbolic polynomials.

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Polynomial Reduction

Polynomial reduction is the division of the polynomial p by the divisor polynomials d1, d2, …, dn . The terms of the divisor polynomials are ordered according to a certain term order. The quotients q1, q2, …, qn and the remainder r satisfy this equation.

$p={q}_{1}{d}_{1}+{q}_{2}{d}_{2}+\dots +{q}_{n}{d}_{n}+r.$

No term in r can be divided by the leading terms of any of the divisors d1, d2, …, dn .