MathLabs

Problem 6

Find all polynomials PP in two variables such that, for a positive integer nn, P(tx,ty)=tnP(x,y)P(tx,ty)=t^nP(x,y) for all real t,x,yt,x,y; for all real a,b,ca,b,c, P(b+c,a)+P(c+a,b)+P(a+b,c)=0P(b+c,a)+P(c+a,b)+P(a+b,c)=0; and P(1,0)=1P(1,0)=1.
Step 3 of 6: Derive an invariance for QQ
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P(x,y)=(x−2y)Q(x,y),deg⁡Q=n−1P(x,y)=(x-2y)Q(x,y),\quad \deg Q=n-1
Detailed analysis

Write P=(x−2y)QP=(x-2y)Q. Substituting (a,b,c)=(a,b,b)(a,b,c)=(a,b,b) into the cyclic identity gives 2(a−b)[Q(a+b,b)−Q(2b,a)]=02(a-b)[Q(a+b,b)-Q(2b,a)]=0, hence Q(x,y)=Q(2y,x−y)Q(x,y)=Q(2y,x-y) for all x,yx,y.