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Polynomials with roots in arithmetic progression.

Source: Spain Mathematical Olympiad 2020 P1

July 14, 2020
algebrapolynomialarithmetic sequenceSpain

Problem Statement

A polynomial p(x)p(x) with real coefficients is said to be almeriense if it is of the form:
p(x)=x3+ax2+bx+a p(x) = x^3+ax^2+bx+a
And its three roots are positive real numbers in arithmetic progression. Find all almeriense polynomials such that p(74)=0p\left(\frac{7}{4}\right) = 0
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