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PolynomialQ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

PolynomialQ[expr, var]
    gives True if expr is a polynomial in var, False otherwise.
PolynomialQ[expr, {v1, v2, ...}]
    gives True if expr is a polynomial in all of the vi simultaneously.
Checks that expr expands to a sum of products of non-negative integer
powers of the vars with var-free coefficients.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= PolynomialQ[x^3 - 2x/y + 3x z, x]
Out[1]= True

In[2]:= PolynomialQ[x^3 - 2x/y + 3x z, y]
Out[2]= False

In[3]:= PolynomialQ[x^2 + a x y^2 - b Sin[c], {x, y}]
Out[3]= True

In[4]:= PolynomialQ[f[a] + f[a]^2, f[a]]
Out[4]= True

Implementation notes

Algorithm. builtin_polynomialq is a structural predicate. It normalises the second argument into a variable set (a single symbol or a List of symbols) and calls is_polynomial, which recurses over the expression tree: an expression is a polynomial in the given variables iff every node is one of the variables, a sub-expression free of all the variables (a degree-0 constant), a Plus/Times whose arguments are all polynomials, or a Power[base, k] with a non-negative integer exponent k and polynomial base. Anything else containing a variable in a non-polynomial position (e.g. Sin[x], 1/x, x^(1/2)) returns False. Returns the symbol True or False.

  • Protected.
  • Variables can be symbols or compound expressions.
  • Constants (expressions free of the specified variables) are polynomials of degree 0.
  • Power[base, exp] is a polynomial if exp is a non-negative integer and base is a polynomial.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= PolynomialQ[x^3 + 2 x + 1, x]
Out[1]= True

In[1]:= PolynomialQ[Sin[x] + x, x]
Out[1]= False

In[1]:= PolynomialQ[x^2 y + x y^2 + 1, {x, y}]
Out[1]= True

In[1]:= PolynomialQ[x^2 + y/x, x]
Out[1]= False

Notes

PolynomialQ[expr, var] tests whether expr is a polynomial in var — i.e. expands to a sum of products of non-negative integer powers of the variables with variable-free coefficients. Sin[x] + x fails because of the transcendental term, and x^2 + y/x fails because y/x = y x^(-1) carries a negative power of x. The multivariate form PolynomialQ[expr, {x, y}] requires polynomiality in all listed variables simultaneously. Note that the test is syntactic up to expansion and does not cancel rational expressions: a fraction such as (x^2 - 1)/(x - 1) is reported False even though it equals the polynomial x + 1.