Some days ago, I've made this question and I guess I've finally found an answer to this question:
- (a) Is it possible to find a polynomial, apart from the constant $0$ itself, which is identically equal to $0$ (i.e. a polynomial $P(t)$ with some nonzero coefficient such that $P(c)=0$ for each number $c$)?
Which is mentioned in that post, I tried to proceed in the following way: I decomposed the polynomial in 2 kinds of structures - due to their similar behaviors inside the polynomial - for example:
The red structures are the $t$'s (1) with a coefficient and the green structure is a number alone, for the red structures, it's impossible to have $P(c)=0$ for every number $c$, with nonzero coefficients. Then I eliminated the reds and started to think about the green alone and I perceived that the only number which would allow me to find this polynomial was zero, thus making it impossible to find the requested polynomial.
Is this valid?
(1) - I have no idea on how to call the $t$'s in the polynomial, the book I'm reading provided me with some terms like: Leading coefficient, coefficients, leading term, constant term/constant coefficient, linear coefficient and linear term, but it mentions no name for the $t$'s, can you provide me a name for it?