MathLabs

Problem 5

Find all functions f:R→Rf:\mathbb R\to\mathbb R such that for all x,y,z∈Rx,y,z\in\mathbb R, f(f(x)+f(y)+f(z))=f(f(x)−f(y))+f(2xy+f(z))+2f(xz−yz)f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).
Step 2 of 8: First symmetry
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f(f(t)−f(0))=f(f(0)−f(t))f(f(t)-f(0))=f(f(0)-f(t))
Detailed analysis

Substituting (t,0,0)(t,0,0) and (0,t,0)(0,t,0) and comparing gives the displayed equality for every real tt.