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Saturday, June 13, 2015

Indian Statistical Institute B.Math & B.Stat : Polynomials

Indian Statistical Institute B.Math & B.Stat Solved Problems, Vinod Singh ~ Kolkata If a polynomial P with integer coefficients has three distinct integer zeroes, then show that P(n)1 for any integer.
Let α,β and γ be the distinct integer zeroes of the polynomial P. If possible let P(m)=1 where mZ. Since PZ(x) we have αm|P(α)P(m)αm|(1). Similarly βm|(1) and γm|(1). Since α,β and γ are distinct αm,βm and γm are distinct. This shows that αm,βm and γm are distinct factors of 1, which is impossible! So the assumption that P(m)=1 is not tenable for any integer m.

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