### 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 |a1a2|<є and the root a verifies a1aa2.
• 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 x3x2−2x+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 x3x2−2x+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]]