<s>[INST] How did people come up with the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$? Every resource that I've read proves the formula
$$ a^3 + b^3 = (a+b)(a^2-ab+b^2) \tag1$$
by just multiplying $(a+b)$ and $(a^2 - ab + b^2)$.
But how did people come up with that formula? Did they think like, "Oh, let's just multiply these polynomials, I don't know why, let's just do it." I don't think that people just pointed a finger at the sky and came up with that formula.
So, how to prove $(1)$?
 [/INST] As noticed by $a=-b$ we obtain $a^3+b^3=0$ therefore we can guess by homogeneity
$$a^3+b^3=(a+b)(Xa^2+Yab+Zb^2)$$
with $X$, $Y$ and $Z$ unknown, to obtain
$$(a+b)(Xa^2+Yab+Zb^2)=Xa^3+Ya^2b+Zab^2+Xa^2b+Yab^2+Zb^3=$$
$$=Xa^3+(Y+X)a^2b+(Z+Y)ab^2+Zb^3$$
which requires

*

*$X=1$

*$Z=1$

*$Y+X=0 \implies Y=-1$

*$Z+Y=0 \implies Y=-1$
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