In this post, it is suggested in comments and answers that a non-zero polynomial of degree $d$ having at most $d$ roots can show that a zero-polynomial must have all zero coefficients. However, I am not seeing how to use this to show that all the coefficients are zero in a zero polynomial, as this claim gives us information about a non-zero polynomial, and we want to show something of the zero polynomial. It seems this claim can only help us show the converse of what we want to show- that a polynomial with all zero coefficients is a zero-polynomial, as if it weren't it would have infinite degree by our claim. So how can we use the fact that a non-zero polynomial of degree $d$ has at most $d$ roots to prove that a zero polynomial must have coefficients all equal to $0$?
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A polynomial p(x) of degree n p(x)=0 for all x, then there mast be at least n+1 xi with p(xi)=0 so one can write it as Product $px)=a(x-x_1)(x-x_2)...(x-x_n)\cdot (x-x_{n+1})$ a polynomial of degree n+1. So there is no p(x)=0 of degree n for any n except a=0 so all coefficients are 0 .
trula
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How do we reason that there is no other way to write this degree $n$ polynomial which doesn't have zero coefficients? I see that writing it this way in a product implies that there exists a representation with all $0$ coefficients – Princess Mia Sep 16 '23 at 23:04
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If it has n+1 zeros one always can write it this way. So iI don't understand your question. – trula Sep 17 '23 at 10:37