# 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)?

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No, it has constants: those are 1 * x^2 + 1 * x + 0 and 1 * x + 1 * x + 0. The constants are the copies of 1 and 0. –  Qiaochu Yuan Jan 30 '11 at 15:30
Just a variable or a constant would also be a polynomial. To parse the Wikipedia statement more clearly, one specifies which variables and which constants (coefficients) are allowed. Then the polynomials are all finite expressions that can be constructed, starting from those variables and constants, and allowing multiplication or addition of two polynomials to again be a polynomial. Such a definition means the variables and constants are included themselves as polynomials. –  hardmath Jan 30 '11 at 17:24
$0$ is also a polynomial. :-) –  a.r. Jan 30 '11 at 18:43

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.
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.