MathDB

Let

P_1(x), P_2(x), \ldots, P_n(x)

be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials A_r(x),B_r(x) (r \equal{} 1, 2, 3) such that \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2 \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2 \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2

Problem Statement