Another question about Hardy's A Course of Pure Mathematics

[MaxMath]

Why "therefore ..." (highlighted in the attached)? It's on page 78 of the pdf. My brain is stuck here. Thanks!

 

[Dr.Peterson]

A

Why "therefore ..." (highlighted in the attached)? It's on page 78 of the pdf. My brain is stuck here. Thanks!

A non-zero polynomial has only a finite number of zeros. Since this polynomial is zero for an infinite number of values (all integer multiples of a), it must be the zero polynomial.

 

[MaxMath]

A non-zero polynomial has only a finite number of zeros. Since this polynomial is zero for an infinite number of values (all integer multiples of a), it must be the zero polynomial.

Thank you. I thought hard and thought this is probably the reason. Before this point in the book Hardy talked about roots of polynomials and their ‘solvability’ which I think supports this. But only a word “therefore” here is really too big a gap for me to fill. Anyway I think that’s it. Appreciate you looking at it.

 

[Steven G]

Thank you. I thought hard and thought this is probably the reason. Before this point in the book Hardy talked about roots of polynomials and their ‘solvability’ which I think supports this. But only a word “therefore” here is really too big a gap for me to fill. Anyway I think that’s it. Appreciate you looking at it.

For me when I saw the words as easily seen, I knew what I was going to be doing that night

 

[MaxMath]

For me when I saw the words as easily seen, I knew what I was going to be doing that night

To me, these words only hurt!

 

[MaxMath]

By intuition, not as a proper proof, this should be true if looking from the other end (not proof by contradiction). When x gets big, the dominant term of a polynomial is the one with the highest order; the other terms effectively do not matter. So, polynomials cannot be periodic when x is big enough; that means the same as polynomials must not be periodic (without a condition).