1. The problem statement, all variables and given/known data Prove that if a0+a1x+a2x2+a3x3+....+anxn=0 then a0=0, a1=0 ... an=0 2. Relevant equations none 3. The attempt at a solution I think I can do this for n up to 2 in the following way (please tell me if you see any gaps in my logic here): f(x)=a0+a1x+a2x2=0 (from the statement above) f(0)=a0+a1(0)+a2(0)2=a0=0 f(1)=(0)+a1(1)+a2(1)2=a1+a2=0 f(-1)=(0)+a1(-1)+a2(-1)2=-a1+a2=0 here it can be easily shown that a1 must be the zero vector for reals, 0, using properties of the zero vector if a0=0 and a1=0 the we have: f(1)=(0)+(0)(1)+a2(1)2=a2=0 I am really not sure where to go from here. At least at first glance, I can't seem to make this same process work for n=3 and even if I could use this process on higher n I'm not sure how/if I could use it to prove the general case for all n. Any help/prodding in the right direction would be much appreciated. Thanks.