Alternative forms

An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For instance (x + 1)3

is a polynomial because it can be worked out to x3 + 3x2 + 3x + 1. Similarly

is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to x^3/12 and 1/12 is just a constant.The coefficient of this term is therefore 1/12.For similar reasons, if complex coefficients are allowed, one may have a single term like (2 + 3i)x3; even though it looks like it should be worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.

Division by an expression containing a variable is not generally allowed in polynomials.[1] For example,

is not a polynomial because it includes division by a variable. Similarly,

is not a polynomial, because it has a variable exponent.

Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.