1=(P+Q+R)^3=P^3+3P^2(Q+R)+3P(Q+R)^2+(Q+R)^3=
P^3+3P^2Q+3P^2R+3PQ^2+6PQR+3PR^2+Q^3+3Q^2R+3QR^2+R^3=
(P^3+Q^3+R^3)+3P(PQ+QR+PR)+3Q(PQ+QR+PR)+3R(PQ+QR+PR)-3PQR.
So (P^3+Q^3+R^3)+3(P+Q+R)(PQ+QR+PR)-3PQR=1
This evaluates to (P^3+Q^3+R^3)+(3*1*-1)+3=1, and (P^3+Q^3+R^3)=1.