content - Maple Help

content

content of a multivariate polynomial

primpart

primitive part of a multivariate polynomial

 Calling Sequence content(a, x, 'pp') primpart(a, x, 'co')

Parameters

 a - multivariate polynomial in x x - (optional) name or set or list of names pp - (optional) unevaluated name co - (optional) unevaluated name

Description

 • If a is a multivariate polynomial, content returns the content of a with respect to x, thus returning the greatest common divisor of the coefficients of a with respect to the indeterminate(s) x.
 • The indeterminate(s) x can be a name, or a list or set of names. If x is not specified, then its default value is the set of all indeterminates occurring in a, as determined by indets.
 • The third argument pp, if present, will be assigned the primitive part of a, namely a divided by the content of a.  For all inputs, the primitive part is a polynomial with integer or floating-point coefficients whose content is 1.
 • The coefficients in a can be arbitrary expressions independent of x. If a is not a polynomial with numeric coefficients, then normal will be called on it (or parts of it), and the numerator and denominator are dealt with separately. Any floating point values are left untouched: any floating-point value is assumed to be indivisible with respect to any other value. Other nonpolynomial, nonrational subexpressions get the same treatment.
 • The sign is removed from the content, and not removed from the primitive part.
 • The primpart command returns a/content(a, x). The third argument co, if present, will be assigned the content.

Examples

The content of a univariate integer polynomial is the GCD of its coefficients.

 > $\mathrm{content}\left(3-3x,x\right)$
 ${3}$ (1)

The content of a multivariate polynomial a with respect to some of its variable(s) x is the GCD of its coefficients, considering a as a polynomial in the variable(s) x with any remaining variables being part of the coefficient ring. In the example below, a is viewed as a polynomial in x with coefficients that are polynomials in y. The example after that takes the same polynomial, but views it as a multivariate polynomial in x and y with integer coefficients.

 > $\mathrm{content}\left(3xy+6{y}^{2},x\right)$
 ${3}{}{y}$ (2)
 > $\mathrm{content}\left(3xy+6{y}^{2},\left[x,y\right]\right)$
 ${3}$ (3)

The following example computes not just the content, but also the primitive part.

 > $\mathrm{content}\left(-4xy+6{y}^{2},x,'\mathrm{pp}'\right)$
 ${2}{}{y}$ (4)
 > $\mathrm{pp}$
 ${3}{}{y}{-}{2}{}{x}$ (5)

In this example, you can see the effect of calling normal, which happens because the polynomial doesn't have purely numeric coefficients (the coefficient of x is $\frac{1}{a}$).

 > $\mathrm{content}\left(\frac{x}{a}-\frac{1}{2},x,'\mathrm{pp}'\right)$
 $\frac{{1}}{{2}{}{a}}$ (6)
 > $\mathrm{pp}$
 ${2}{}{x}{-}{a}$ (7)
 > $\mathrm{normal}\left(\frac{x}{a}-\frac{1}{2}\right)$
 ${-}\frac{{-}{2}{}{x}{+}{a}}{{2}{}{a}}$ (8)

Floating point coefficients are considered indivisible with respect to each other -- even if they are equal. As a consequence, the content in the following example is 1.

 > $\mathrm{content}\left(2.ux-2.v,x,'\mathrm{pp}'\right)$
 ${1}$ (9)
 > $\mathrm{pp}$
 ${2.}{}{u}{}{x}{-}{2.}{}{v}$ (10)

In the presence of floating-point numbers, other content is still detected. For example, the factor u below.

 > $\mathrm{content}\left(2.ux-2.u,x,'\mathrm{pp}'\right)$
 ${u}$ (11)
 > $\mathrm{pp}$
 ${2.}{}{x}{-}{2.}$ (12)

Non-numeric, nonpolynomial coefficients are also considered indivisible with respect to each other. For example, you could consider $\sqrt{2}$ to be a common divisor between the two coefficients $\sqrt{10}$ and $\sqrt{6}$, but they are considered indivisible with respect to each other for this command and the content is considered to be 1.

 > $\mathrm{content}\left(\mathrm{sqrt}\left(10\right)x+\mathrm{sqrt}\left(6\right),x\right)$
 ${1}$ (13)

The primpart command computes just the primitive part of the expression.

 > $\mathrm{primpart}\left(-4xy+6{y}^{2},x\right)$
 ${3}{}{y}{-}{2}{}{x}$ (14)
 > $\mathrm{primpart}\left(\frac{x}{a}-\frac{1}{2},x\right)$
 ${2}{}{x}{-}{a}$ (15)

Compatibility

 • The a parameter was updated in Maple 2017.