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integer f(x) = x^4 +ax^3 +bx^2 +cx+d with exactly one real root

Source: Israel Grosman Memorial Mathematical Olympiad 1999 p4

February 15, 2020

Integer PolynomialpolynomialFactoringalgebra

Problem Statement

Consider a polynomial

f(x) = x^4 +ax^3 +bx^2 +cx+d

with integer coefficients. Prove that if

f(x)

has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients