<s>[INST] How to solve this algebraic manipulation problem? Express the following expression $$E=(x^3-y^3)(y^3-z^3)(z^3-x^3)$$ in terms of $a, b$ where $a,b \in \mathbb R$ and 
$$a=x^2y+y^2z+z^2x$$ $$b=xy^2+yz^2+zx^2$$
 [/INST] Firstly, we use $x^3-y^3=(x-y)(x^2+xy+y^2)$ for $(x,y)=(a,b),(b,c),(a,c)$ and we obtain
$$E=(x-y)(y-z)(z-x)(x^2+xy+y^2)(y^2+yz+z^2)(x^2+xz+z^2)$$
$$E=(x^2z+y^2x+z^2y-x^2y-y^2z-z^2x)(x^2+xy+y^2)(y^2+yz+z^2)(x^2+xz+z^2)$$
$$E=(b-a)(x^2+xy+y^2)(y^2+yz+z^2)(x^2+xz+z^2)=(b-a)E^{'}$$
$$\text{Let's say }c=\frac{a}{xyz}=\frac{x}{z} + \frac {y}{x} + \frac {z}{y} \text{and  }d=\frac{b}{xyz}=\frac{x}{y} + \frac {y}{z} + \frac {z}{x}$$
$$\frac{E^{'}}{(xyz)^2}=\prod_{cyc}{(x/yz+1/z+y/xz)}=(\sum_{cyc} \frac{x^2}{y^2}  + 2 * \sum_{cyc} \frac{x}{z}) + ( \sum_{cyc} \frac{x^2}{z^2} + 2* \sum_{cyc} \frac{x}{y} ) + (\sum {xy/z^2} +3 ) $$
$$\text{Therefore, }\frac{E^{'}}{(xyz)^2}=c^2 + d^2 + cd=\frac{a^2+b^2+ab}{(xyz)^2}$$
$$\text{Now we have obtained }E^{'}=a^2+b^2+ab\text{, and we know, from earlier, }E=(b-a)E^{'}\text{, therefore}$$
$$E=(b-a)(a^2+b^2+ab) $$
$$E=b^3-a^3 $$
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