### 2.29.2 Exact bounds for real roots of a polynomial : `realroot`

`realroot` has 2 or 4 arguments : a polynomial and a real number є and optionnally two reals numbers α,β.
`realroot` returns a list of vectors.

- If `realroot` has 2 arguments, the elements of each vector are
  - either a real interval containing a real root of the polynomial and the multiplicity of this root. Let the interval be [a1,a2] then |a1−a2|<є and the root a verifies a1≤ a ≤ a2.
  - or the value of an exact real root of the polynomial and the multiplicity of this roots
- If `realroot` has 4 arguments, `realroot` returns a list of vectors as above, but only for the roots inside the interval [α,β].

To find the real roots of x3+1, input:

`realroot(x```^```3+1, 0.1)`

Output :

`[[-1,1]]`

To find the real roots of x3−x2−2x+2, input:

`realroot(x```^```3-x```^```2-2*x+2, 0.1)`

Output :

`[[1,1],[[(-3)/2,(-45)/32],1],[[45/32,3/2],1]]`

To find the real roots of x3−x2−2x+2 in the interval [0;2], input:

`realroot(x```^```3-x```^```2-2*x+2, 0.1,0,2)`

Output :

`[[1,1],[[11/8,23/16],1]]`