# How to simplify an expression with no constraints?

Simplifying expressions on Mathematica is a recurrent topic. However I found myself stuck with this one:

Solving an rational equation on Mathematica gives me the following expression:

expression := -((2 x)/3) + (2^(1/3) (3 - x^2))/(
3 (9 x - 2 x^3 + Sqrt[4 (3 - x^2)^3 + (9 x - 2 x^3)^2])^(
1/3)) - (9 x - 2 x^3 + Sqrt[4 (3 - x^2)^3 + (9 x - 2 x^3)^2])^(
1/3)/(3 2^(1/3))


I know that the expression should simply to $$-x$$. If I plot the expression I get indeed a plot of $$y=-x$$.

Plot[expression, {x, -1, 1}] I tried using the function "Simplify", "FullSimplify" even "Expand" and then "Simplify" but got no luck as I do not have restriction on $$x$$. The expression is very messy and I do not manage to simplify it by hand. How could I use mathematica to simplify "expression" to $$-x$$? (other than using a graphical plot)

Any idea are much appreciated! Thanks

• Try: Plot[expression, {x, -10, 2}] ? Aug 8, 2019 at 17:25
• How much do you trust magic? In:= First[GroebnerBasis[expression, x]] Out= x Jan 6, 2020 at 14:57

This isn't the best or most general solution, but you can experiment a bit by Taylor expanding about $$x=0$$ to some sufficiently high order.
Series[expression, {x, 0, 20}] // Simplify
This results in -x+O[x]^21 which means there are no corrections to $$-x$$ upto order 21 in the Taylor expansion about $$x=0$$. If this remains stable to even higher orders you have a probable guess as to what the result is.
• As @Mariusz Iwaniuk points out the plot isn't a straight line for $x<-2$. This happens clearly because there are branch cuts at $x=\pm2$ from the $\sqrt{4-x^2}$ appearing in the expression. This is obviously not seen if you do a Taylor expansion about $x=0$. Aug 8, 2019 at 19:04