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2 polynomias of 1998 degree

Source: 1998 Romania NMO VIII p2

August 14, 2024
algebrapolynomial

Problem Statement

Let P(x)=a1998X1998+a1997X1997+...+a1X+a0P(x) = a_{1998}X^{1998} + a_{1997}X^{1997} +...+a_1X + a_0 be a polynomial with real coefficients such that P(0)≠P(−1)P(0) \ne P (-1), and let a,ba, b be real numbers. Let Q(x)=b1998X1998+b1997X1997+...+b1X+b0Q(x) = b_{1998}X^{1998} + b_{1997}X^{1997} +...+b_1X + b_0 be the polynomial with real coefficients obtained by taking bk=aak+bb_k = aa_k + b ,∀k=0,1,2,...,1998\forall k = 0, 1,2,..., 1998. Show that if Q(0)=Q(−1)≠0Q(0) = Q (-1) \ne 0 , then the polynomial QQ has no real roots.
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