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 5 of 6: Use homogeneity and normalization
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Q(x,y)=Q(x+y,0)=Q(1,0)(x+y)n−1=(x+y)n−1Q(x,y)=Q(x+y,0)=Q(1,0)(x+y)^{n-1}=(x+y)^{n-1}
Detailed analysis

Take d=yd=y in Step 4: Q(x,y)=Q(x+y,0)Q(x,y)=Q(x+y,0). Homogeneity gives Q(x+y,0)=Q(1,0)(x+y)n−1Q(x+y,0)=Q(1,0)(x+y)^{n-1}. Since P(1,0)=Q(1,0)=1P(1,0)=Q(1,0)=1, we obtain Q=(x+y)n−1Q=(x+y)^{n-1} and hence P=(x−2y)(x+y)n−1P=(x-2y)(x+y)^{n-1}.