| Version 6 |
Version 5 |
| The {\em zero polynomial} in a ring $R[X]$ of polynomials over a ring $R$ is the additive identity element $\textbf{0}$ of this polynomial ring: |
The {\em zero polynomial} in a ring $R[X]$ of polynomials over a ring $R$ is the additive identity element $\textbf{0}$ of this polynomial ring: |
| $$f+\textbf{0} = \textbf{0}+f = f \quad\forall\, f\in R[X]$$ |
$$f+\textbf{0} = \textbf{0}+f = f \quad\forall\, f\in R[X]$$ |
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| All coefficients of the zero polynomial are equal to 0, i.e. |
All coefficients of the zero polynomial are equal to 0, i.e. |
| $$\textbf{0} := (0,\,0,\,0,\,...).$$ |
$$\textbf{0} := (0,\,0,\,0,\,...).$$ |
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| Because always |
Because always |
| $$f\cdot\textbf{0} = \textbf{0}$$ |
$$f\cdot\textbf{0} = \textbf{0}$$ |
| and because in general \,$\deg(fg) = \deg(f)+\deg(g)$\, when $R$ has no zero divisors, one may define that |
and because in general \,$\deg(fg) = \deg(f)+\deg(g)$\, when $R$ has no zero divisors, one may define that |
| $$\deg(\textbf{0}) = -\infty$$ |
$$\deg(\textbf{0}) = -\infty$$ |
| or that the zero polynomial has no \PMlinkname{degree}{Monomial} at all. |
or that the zero polynomial has no \PMlinkname{degree}{Monomial} at all. |
