Problem 88

Problem
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Week

PROBLEM 88

The following problem was suggested by Philippe Fondanaiche. Merci, Philippe!

A teacher writes a cubic equation on the board:

p(x) = x³ + p x² + q x + r with p,q,r integers. He tells his students that that polynomial has three integer roots, and that the largest of the roots is twice the smallest of the roots.

A student guesses a value, A, for the largest root which is too low, and for which p(A) is positive. A second student guesses a value, B, for the largest root which is too high and for which p(B) again positive. A third student, who was dozing in the back of the classroom, notices that each of the four numbers mentioned, A, B, p(A), p(B), is prime and one of them is the number 13.

What are the roots of p(x)?

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