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 2017-04-25, 04:16 #2 LaurV Romulan Interpreter     Jun 2011 Thailand 22×3×739 Posts Going to differentiation may be a bit tricky and not exactly "elementary" (as we learn polynomials in school much earlier than differentiation). One simpler way is to express the two in zero, getting $$c=t$$, then express them in 1 and -1 (or 2, etc) and get $$b=s$$, $$a=r$$ etc. We have: $$f(0)=g(0),\ \ \text{i.e.}\ \ a\cdot 0^2+b\cdot 0+c=r\cdot 0^2+s\cdot 0+t$$ from which $$s=t$$, then express them in 1 and eliminate t=c from the both sides:$$f(1)=g(1),\ \ \text{i.e.}\ \ a\cdot 1^2+b\cdot 1+c=r\cdot 0^2+s\cdot 0+t$$ or $$a+b=r+s$$. Doing the same for -1, we get $$a-b=r-s$$ and now we add them together or subtract them together, and get a=r and b=s. This also proves that it is enough for two polynomials of degree n to be the same in n+1 values (as they have n+1 coefficients), to be the same in all their domain. (i.e. if two polynomials of degree n in R have the same values in n+1 points, they are the same in all R, following a similar procedure, and induction). Again, congratulations for your effort and for your patience to write another excellent and educative post! Last fiddled with by LaurV on 2017-04-25 at 04:32 Reason: fixing \tex thingies