Problem
ALG-B3-M07-P020 Triple with Zero Sum
#20
★★★★★ Level 5 of 5
Let \(P\in\mathbb R[x]\) and \(P(x)+P(y)+P(-x-y)=0\) for all \(x,y\). Find \(P\).
Look at the leading homogeneous part and put \(y=x\).
Let the degree of \(P\) be \(d\), with leading coefficient \(a e0\). The leading part of the equality is \(a(x^d+y^d+(-x-y)^d)\). Put \(y=x\): we get \(a(2+(-2)^d)x^d\). If \(d\ge2\), this coefficient is nonzero: for even \(d\) it is positive, and for odd \(d\ge3\) it is \(2-2^d\). Hence \(d\le1\). Let \(P(x)=ax+b\). Substitution gives \(a(x+y-x-y)+3b=0\), so \(b=0\). The answer is \(P(x)=ax\), including \(a=0\).
Final task on the leading part of a polynomial.