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divide

Divide polynomials

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

divide(p, q, <[x]>, <order>, options)
divide(p, q, <[x1, x2, …]>, <order>, options)
divide(p, q1, q2, …, <order>, options)

Description

divide(p, q) divides polynomials or polynomial expressions p and q. By default, the function returns the quotient s and the remainder r, such that p = sq + r. Here degree(r) < degree(q).

divide(p, q1, q2, q3, …, qn) divides a polynomial or a polynomial expression p by polynomials or polynomial expressions q1, q2, q3, …, qn. The function returns the quotients s1, s2, s3, …, sn and the remainder r, such that p = s1q1 + s2q2 + s3q3 + … + s4q4 + r. Here the leading coefficient of the remainder r cannot be divided by the leading coefficients of any of divisors q1, q2, q3, …, qn.

divide(p, q) divides the polynomial or polynomial expression p by the polynomial or polynomial expression q. Use the Quo option to return the quotient only. Use the Rem option to return the remainder only.

The divide function operates on polynomials or polynomial expressions.

Polynomials must be of the same type: their variables and coefficient rings must be identical.

When you call divide for polynomial expressions, MuPAD® internally converts these expressions to polynomials. See the poly function. If you do not specify the list of indeterminates, divide treats all symbolic variables in the expressions as indeterminates. If the expressions cannot be converted to polynomials, the divide function returns FAIL. See Example 1.

If you call divide for polynomials, it returns polynomials. If you call divide for polynomial expressions, the function returns polynomial expressions. See Example 2.

If you divide polynomial expressions that contain more than one variable, you can specify particular variables to be treated as variables. The divide function treats all other variables as symbolic parameters. By default, divide assumes that all variables in polynomial expressions are variables, and none of them is a symbolic parameter. See Example 3.

divide(p, q1, q2, q3, …, qn) divides a polynomial or a polynomial expression p by polynomials or polynomial expressions q1, q2, q3, …, qn. The function returns quotients s1, s2, s3, …, sn and remainder r, such that p = s1q1 + s2q2 + s3q3 + … + s4q4 + r. Here the leading coefficient of the remainder r cannot be divided by the leading coefficients of any of the divisors q1, q2, q3, …, qn. See Example 6.

When dividing a polynomial by one or more polynomials, you can select the term ordering. The ordering accepts the following values:

  • LexOrder sets the lexicographical ordering.

  • DegreeOrder sets the total degree ordering. When using this ordering, MuPAD sorts the terms of a polynomial according to the total degree of each term (the sum of the exponents of the variables).

  • DegInvLexOrder sets the total degree inverse lexicographic ordering. When using this ordering, MuPAD sorts the terms of a polynomial according to the total degree of each term (the sum of the exponents of the variables). If the several terms have equal total degrees, MuPAD sorts them using the inverse lexicographic ordering.

  • your custom term ordering of type Dom::MonomOrdering.

The coefficient ring of the polynomials must implement the "_divide" method. MuPAD uses this method internally to divide coefficients. If the coefficients cannot be divided, this method must return FAIL.

Examples

Example 1

For polynomial expressions, divide internally calls the poly function, which converts an expression to a polynomial. If you do not specify the indeterminate of an expression, MuPAD assumes that all variables are indeterminates. For example, The divide function cannot divide the following polynomial expressions because it assumes that both x and y are indeterminates:

divide(x/y, x)

If you specify that only x is an indeterminate, the result is:

divide(x/y, x, [x])

Example 2

The divide divides polynomials or polynomial expressions. When you divide polynomials, the function returns polynomials:

divide(poly(x^3 + x + 1, [x]), poly(x^2 + x + 1, [x]))

When you divide polynomial expressions, MuPAD internally converts these expressions to polynomials, divides these polynomials, and then converts the result of division to polynomial expressions:

divide(x^3 + x + 1, x^2 + x + 1)

Example 3

When dividing multivariate polynomials, you can specify the list of variables. The divide function assumes all other variables are symbolic parameters. For example, divide the following two polynomial expressions specifying that both x and y are variables:

divide(x^2 - 2*x - y, y*x - 1, [x, y])

Divide the same polynomial expressions specifying that only x is a variable. MuPAD assumes that y is a symbolic parameter:

divide(x^2 - 2*x - y, y*x - 1, [x])

Now, divide these expressions specifying that only y is a variable. MuPAD assumes that x is a symbolic parameter:

divide(x^2 - 2*x - y, y*x - 1, [y])

By default, the divide function treats polynomial expressions with more than one variable as multivariate polynomial expressions. The function does not assume that any of the variables are symbolic parameters:

divide(x^2 - 2*x - y, y*x - 1)

Example 4

By default, divide returns the quotient and the remainder of the division of polynomials:

divide(x^3 + x + 1, x^2 + x + 1)

To return the quotient only, use the Quo option:

divide(x^3 + x + 1, x^2 + x + 1, Quo)

To return the remainder only, use the Rem option:

divide(x^3 + x + 1, x^2 + x + 1, Rem)

Example 5

Suppose, you want to get the result of the division only when the exact division is possible. To return the quotient s of the exact division of polynomials or polynomial expressions, use the Exact option:

divide(x^4 + 12*x^3 + 28*x^2 + 204*x + 187, x + 11, Exact)

When exact division without remainder is impossible, the divide function with the Exact option returns FAIL:

divide(x^4 + 12*x^3 + 28*x^2 + 204*x + 187, x + 12, Exact)

Example 6

The divide function allows you to divide a polynomial (or polynomial expression) by multiple polynomials (or polynomial expressions):

divide(4*x^4 + 2*x^2 + 1, x^3 - x + 1, x - 1)

When dividing a polynomial by multiple polynomials, you can select the term ordering:

divide(x^2+y^3+1, x-y^2, y, LexOrder)

divide(x^2+y^3+1, x-y^2, y, DegreeOrder)

Parameters

p, q

Univariate or multivariate polynomials or polynomial expressions.

p, q1, q2, …

Univariate or multivariate polynomials or polynomial expressions.

x

The indeterminate of the polynomial: typically, an identifier or an indexed identifier. divide treats the expressions as univariate polynomials in the indeterminate x.

x1, x2, …

The indeterminates of the polynomial: typically, identifiers or indexed identifiers. divide treats multivariate expressions as multivariate polynomials in these indeterminates.

order

The term ordering when dividing one multivariate polynomial by one or more multivariate polynomials: LexOrder, DegreeOrder, DegInvLexOrder, or a custom term ordering of type Dom::MonomOrdering. The default is the lexicographical ordering LexOrder.

Options

Exact

Return the quotient s of the exact division of multivariate polynomials. If no exact division without remainder is possible, return FAIL.

Quo, Rem

Return the quotient s or the remainder r. By default, the divide function returns both the quotient and the remainder.

Return Values

Polynomial, a polynomial expression, a sequence of polynomials or polynomial expressions, or the value FAIL.

Overloaded By

p, q

See Also

MuPAD Functions

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