ISL 1989 Problem 56

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 $A_r(x),B_r(x) (r = 1, 2, 3)$ such that

$$\sum^n_{s=1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 + \left( B_1(x) \right)^2$$ $$\sum^n_{s=1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 + x \left( B_2(x) \right)^2$$ $$\sum^n_{s=1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 - x \left( B_3(x) \right)^2$$