# Why is a polynomial with infinite zeropoints the zeropolymomial? [duplicate]

This was given us as a fact, but why is this true? The zeropolynomial is the polynomial where all the coefficients are equal to $$0$$ if $$R(x)$$ is a polynomial over $$\mathbb{C}$$ and every $$x\in \mathbb{N}_0$$ is a zeropoint then one can rewrite the polynomial $$R(x)=x(x-1)(x-2)…(x-n)…$$ but if we put an $$x\in \mathbb{C}-\mathbb{N}_0$$ in $$R(x)$$, how does one know that $$R(x)=0$$?

## marked as duplicate by Dietrich Burde, Community♦Dec 16 '18 at 12:54

A polynomial $$f(x)$$ has by definition a finite degree $$n$$ which is given by the highest degree $$n$$ of the variable $$x$$ involved in the polynomial. If you multiply two polynomials, the degrees add (if the underlying coefficient ring has no zero divisors as in the case of a field). Only the zero polynomial has all elements (in the coefficient ring or an extension ring) as zeros. A polynomial as described cannot exist.
By the Fundamental theorem of algebra, a polynomial $$p$$ of degree $$n>0$$ has exactly $$n$$ complex roots (counting multiplicity). Let $$r_1,\dots,r_n$$ be its root, then we must have $$p(z)\ne 0$$ for all $$z$$ such that $$|z| > \max\{|r_1|,\dots,|r_n|\} =: R$$ since all the roots lie in the set $$\{z\in\Bbb C: |z|\le R\}$$.
This means that the only polynomial with infinitely many zeroes is the consstant polynomial $$p\equiv0$$.