Polynomials, Integers and Divisibility
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Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that
f(n)?f(m) is divisible by n ? m for all distinct integers n and m.
See the attached file.
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Solution Summary
Polynomials, integers and divisibility are investigated and discussed in the solution.
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Ok, the main thing to observe here is that fact that: for any integer like p and distinct integers like m and n, (n^p-m^p) is divisible by (n-m). In fact we always have that:
(n^p-m^p)=(n-m)(n^(p-1)+n^(p-2)*m+n^(p-3)*m^2+...+n^2*m^(p-3)+n*m^(p-2)+m^(p-1))
That can be verified by direct division i.e. ...
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