(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+....+a_{n}x^{n}=0 then a_{0}=0, a_{1}=0 ... a_{n}=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)=a_{0}+a_{1}x+a_{2}x^{2}=0 (from the statement above)

f(0)=a_{0}+a_{1}(0)+a_{2}(0)^{2}=a_{0}=0

f(1)=(0)+a_{1}(1)+a_{2}(1)^{2}=a_{1}+a_{2}=0

f(-1)=(0)+a_{1}(-1)+a_{2}(-1)^{2}=-a_{1}+a_{2}=0

here it can be easily shown that a_{1}must be the zero vector for reals, 0, using properties of the zero vector

if a_{0}=0 and a_{1}=0 the we have:

f(1)=(0)+(0)(1)+a_{2}(1)^{2}=a_{2}=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.

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# Homework Help: Polynomial Proof

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