Prove that $\dfrac{x³+y³+z³}{x+y+z}=x²+y+z$; if $x<y<z$; $y=x+1$; $z=y+1$; and $x$, $y$ and $z$ are positive whole numbers.
If you prove this, I technically discovered a new formula since I couldn't find it anywhere else. I tried it with a lot of values and it worked, but I still don't have a solid proof, so I'm asking for help.
Edit: thanks for the help. I've called it the Abdelganic Cubic Formula.
It's a polynomial equation of degree $3$ so it suffices to test $4$ instances. Let me explain
Once you substitute $y=x+1$ and $z=x+2$ you see that we want to prove that
$$x^3 + (x+1)^3 + (x+2)^3 = (x^2 + (x+1) + (x+2))(x^2 + (x+1)+(x+2))$$
One option is to open up the brackets and see if everything cancels out.
Another option is to use the fact that a polynomial $p(x)$ of degree $3$ has at most $3$ zeroes, unless $p(x)=0$ for all $x$. Therefore, if you show that the equality above is valid for, say $x=2,3,4,5$ then this suffices to prove that it is always valid.