<s>[INST] Proving an algebraic identity using the axioms of field I am trying to prove (based on the axioms of field) that
$$a^3-b^3=(a-b)(a^2+ab+b^2)$$
So, my first thought was to use the distributive law to show that
$$(a-b)(a^2+ab+b^2)=(a-b)\cdot a^2+(a-b)\cdot ab+(a-b)\cdot b^2$$
And then continuing from this point.
My problem is that I'm not sure if the distributive law is enough to prove this identity. Any ideas? Thanks!
 [/INST] Indeed you need distributive, associative and commutative laws to prove your statement.
In fact:
$$\begin{split}
(a-b)(a^2+ab+b^2) &= (a+(-b))a^2 +(a+(-b))ab+(a+(-b))b^2\\
&= a^3 +(- b)a^2+a^2b+(-b)(ab)+ab^2+(-b)b^2\\
&= a^3 - ba^2+a^2b - b(ab) +ab^2-b^3\\
&= a^3 - a^2b+a^2b - (ba)b +ab^2-b^3\\
&= a^3 -(ab)b +ab^2-b^3\\
&= a^3 -ab^2 +ab^2-b^3\\
&= a^3-b^3\; .
\end{split}$$
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