polynomialReduce

Reduce polynomials by division

Syntax

r = polynomialReduce(p,d)

r = polynomialReduce(p,d,vars)

r = polynomialReduce(___,'MonomialOrder',MonomialOrder)

[r,q] = polynomialReduce(___)

Description

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.

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

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'.

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

Examples

Divide Two Polynomials

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 Polynomial Variables

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 Polynomial by Several Polynomials

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

Specify Term Order of Polynomials

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

p — Polynomial to divide

symbolic expression | symbolic function

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

d — Polynomials to divide by

symbolic expression or function | vector of symbolic expressions or functions

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

vars — Polynomial variables

vector of symbolic variables

Polynomial variables, specified as a vector of symbolic variables.

MonomialOrder — Monomial order of divisors

'degreeInverseLexicographic' (default) | 'degreeLexicographic' | 'lexicographic'

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

r — Remainder of polynomial division

symbolic polynomial

Remainder of polynomial division, returned as a symbolic polynomial.

q — Quotient of polynomial division

symbolic polynomial | vector of symbolic polynomials

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

More About

Polynomial Reduction

Polynomial reduction is the division of the polynomial p by the divisor polynomials d₁, d₂, …, d_n . The terms of the divisor polynomials are ordered according to a certain term order. The quotients q₁, q₂, …, q_n 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, …, d_n .

Version History

Introduced in R2018a