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

Source: Polish MO Second Round 1975 p1

September 8, 2024
algebrapolynomial

Problem Statement

The polynomial W(x)=x4+ax3+bx+cx+d W(x) = x^4 + ax^3 + bx + cx + d is given. Prove that if the equation W(x)=0 W(x) = 0 has four real roots, then for there to exist m m such that W(x+m)=x4+px2+q W(x+m) = x^4+px^2+q , it is necessary and it is enough that the sum of certain two roots of the equation W(x)=0 W(x) = 0 equals the sum of the remaining ones.
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