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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
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$0$ is also a polynomial. :-) –  a.r. Jan 30 '11 at 18:43
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2 Answers

up vote 4 down vote accepted

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.

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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.

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so "0" or even an empty expression are polynomials? –  Tom Brito Feb 16 '11 at 11:26
    
@Tom Brito: “0” indeed is a polynomial, the zero polynomial it is called, one of constant polynomials.The empty expression is not a polynomial because it is not of the desired form (is not well-formed). –  beroal Feb 19 '11 at 5:02
    
@Tom Brito: Your textbook should contain the definition of the well-formed expressions <strong>in the ring</strong>. If it does not, it suppose that you know it intuitively. The definition is not long, but I am reluctant to repeat it again. Search for the keywords: “term”, “functional symbol”, “algebraic structure”. –  beroal Feb 19 '11 at 5:14
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