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An interestin polynomial problem with integer roots

Source: Austrian Mathematical Olympiad 1998, Part 2, D2, P2

June 29, 2011
algebrapolynomialgreatest common divisornumber theory proposednumber theory

Problem Statement

Let P(x)=x3−px2+qx−rP(x) = x^3 - px^2 + qx - r be a cubic polynomial with integer roots a,b,ca, b, c.
(a) Show that the greatest common divisor of p,q,rp, q, r is equal to 11 if the greatest common divisor of a,b,ca, b, c is equal to 11.
(b) What are the roots of polynomial Q(x)=x3−98x2+98sx−98tQ(x) = x^3-98x^2+98sx-98t with s,ts, t positive integers.
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