Question: What does "nonzero" polynomial mean?
What does "nonzero" polynomial mean?
Thank you
$\endgroup$ 51 Answer
$\begingroup$Since there is no answer, here is what I stated in the comments:
Usually, a nonzero polynomial $f$ is a polynomial of where not every coefficient is zero, i.e. $$f(X)=\sum\limits_{k=0}^n a_kX^k\quad(n\ge0)$$ and one of the $a_i\neq 0$.
Depending on context, even the definition that $f(x)\neq0$ for some $x$ could be used, however this is rare.
It might seem as if these were equivalent, however consider $$f(X)=X^2+X$$ in $\mathbb{Z}/2\mathbb{Z}$. For $0$, it obviously evaluates to zero, for $1$ it is $f(1)=1^2+1=1+1=0$ (because $1+1=0$ in $\mathbb{Z}/2\mathbb{Z}$).
This polynomial is zero regarding the second definition, not regarding the first definition however.
Therefore, the first definition is almost always used.
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