$\begingroup$How to prove that every polynomial with real coefficients is the sum of three polynomials raised to the 3rd degree? Formally the statement is:
$\endgroup$6$\forall f\in\mathbb{R}[x]\quad \exists g,h,p\in\mathbb{R}[x]\quad f=g^3+h^3+p^3$
1 Answer
$\begingroup$We have that the following identity holds $$(x+1)^3+2(-x)^3+(x-1)^3=6x.$$ Hence $$\left(\frac{f(x)+1}{6^{1/3}}\right)^{3}+\left(\frac{-f(x)}{3^{1/3}}\right)^{3}+ \left(\frac{f(x)-1}{6^{1/3}}\right)^{3}=f(x).$$
$\endgroup$7
