algebra precalculus - The Polynomial concept needs to include both variables and contants?

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

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.