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Prove that P(x+1)=Q(x-1) has a real solution if P(x)=Q(x) has no real solution

Source: MTRP 2017 Class 11-Short Answer Type Question: Problem 1 :-

May 26, 2020

algebrapolynomialReal Rootsmonic

Problem Statement

A monic polynomial is a polynomial whose highest degree coefficient is 1. Let

P(x)

and

Q(x)

be monic polynomial with real coefficients and

degP(x)=degQ(x)=10

. Prove that if the equation

P(x)=Q(x)

has no real solutions then

P(x+1)=Q(x-1)

has a real solution