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polynomials L of deg(L)=3,4

Source: Romanian TST 1978, Day 1, P7

September 28, 2018
algebrapolynomial

Problem Statement

Let P,Q,R P,Q,R be polynomials of degree 3 3 with real coefficients such that P(x)≤Q(x)≤R(x), P(x)\le Q(x)\le R(x) , for every real x. x. Suppose P−R P-R admits a root. Show that Q=kP+(1−k)R, Q=kP+(1-k)R, for some real number k∈[0,1]. k\in [0,1] . What happens if P,Q,R P,Q,R are of degree 4, 4, under the same circumstances?
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