Problem 6

Find all polynomials

P

in two variables such that, for a positive integer

n

P(tx,ty)=t^nP(x,y)

for all real

t,x,y

; for all real

a,b,c

P(b+c,a)+P(c+a,b)+P(a+b,c)=0

; and

P(1,0)=1

Step 6 of 6: Verify the family and finish

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P(x,y)=(x-2y)(x+y)^{n-1}

Detailed analysis

This polynomial is homogeneous of degree $n$ and has $P(1,0)=1$ . In the cyclic sum, put $s=a+b+c$ : each term equals $(s-3a)s^{n-1}$ , $(s-3b)s^{n-1}$ , or $(s-3c)s^{n-1}$ ; their sum is zero. Thus these, and only these, are the solutions.