The Polynomial concept needs to include both variables and contants?

Question

As in Wikipedia:

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants.

So, it's only considered a polynomial if it has both variables and constants?

$x^2 + x$ or $x + x$ are not polynomials (as they only have variables)?

Answer 1

As Qiaochu pointed out, the coefficients of the polynomials are the constants. Hence, in a polynomial like $x^2 + x + 5$, the constants are: $1$ for $x^2$, $1$ for $x^1 = x$ and $5$ for $x^0 = 1$. $3$ constants for the $3$ terms.

Answer 2

I'd like to give an answer which is different from the accepted one. If you consider “expression” and “term” being synonyms, then $x\cdot x+x$ does not include constants. But it is still a polynomial because your definition does not say “at least one constant”. Compare to this example: if “a list is a sequence of elements”, the list may contain no elements as well.