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not all the roots of ax^4+bx^3+x^2+x+1=0 can be real when a \ne 0

Source: CRMO 2012 Region 1 p6

September 30, 2018

polynomialpolynomial equationReal Rootsalgebra

Problem Statement

Let

a

and

b

be real numbers such that

a \ne 0

. Prove that not all the roots of

ax^4 + bx^3 + x^2 + x + 1 = 0

can be real.