Polynomial P equals to 0

Question

I have simple questions about polynomials (when I say "polynomials", I mean "formal polynomials", not polynomial mappings).

It might be a little bit strange, but I don't really understand why if we have a polynomial $P = a_{0} + a_{1}X + a_{2}X^{2} + ... + a_{n}X^{n} \in \mathbb{R}[X]$ (where $n \in \mathbb{N}^{\ast}$), then : $$P = 0_{\mathbb{R}[X]} \Rightarrow \forall i \in \{1, ..., n\}, a_{i} = 0$$

Also, if I write $P = a_{0} + a_{1}X + a_{2}'X^{2} + a_{2}X^{2} + ... + a_{n}X^{n}$, can I conclude that : $$P = 0_{\mathbb{R}[X]} \Rightarrow \forall i \in \{1, ..., n\}, a_{i} = 0 \wedge a_{2}' = 0$$ or just that : $$P = 0_{\mathbb{R}[X]} \Rightarrow \forall i \in \{1, ..., n\} - \{2\}, a_{i} = 0 \wedge a_{2} + a_{2}' = 0?$$

Thank you for your answers.

Answer 1

The polynomial $P$ has at most $n$ roots, unless it is equal to the $0_{\Bbb{R[X]}}$ . The last by definition means $a_i=0$. That $P$ has infinite number of roots, we conclude that $a_i=0$ .

About the second question we may conclude that $a_2+a'_2=0$ .

Answer 2

Do you know when are two polynomials the same? Say we have $$p(x)=a_nx^n+...+a_2x^2+a_1x+a_0$$ and

$$q(x)=b_mx^m+...+b_2x^2+b_1x+b_0$$

then $p(x)= q(x) $ iff $m=n$ and $$a_0=b_0;$$ $$a_1=b_1;$$ $$a_2=b_2;$$ $$\vdots $$ $$a_n=b_n;$$

So if $q(x)=0$ then $a_0=a_1=a_2=...=0$ and you can not say $a_2=a_2'=0$. You can and must say $a_2+a_2'=0$.