Given a polynomial equation (with real coefficients of any degree with any number of repeating roots), let say
$x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24 = 0$, ... (A)
it can be written as
$(x-1)^2 (x+8) (x^2-3) = 0$.
As you can see, it has a repeating root $x = 1$.
Forming a new polynomial by removing the duplicate root (i.e, taking repeating roots only once), we will get
$(x-1) (x+8) (x^2-3) = 0$.
Then expanding it, we get
$x^4+7 x^3-11 x^2-21 x+24 = 0$. ... (B)
Is it possible to get a polynomial having all of the roots of the given polynomial but taking the repeating roots only once, i.e, getting (B) from (A), directly without finding any roots? If possible, how to do the same?