# Why, in any equation, are repeated solutions given by Sage only once?

In the example

var('f, q, r, h, i, g, x, y, z')
eq1 = x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1
solve(eq1==0,x)


Where, this equation is also equal to (x-1)^3*(x+1)^2, but I obtain the following solution:

[x == -1, x == 1]


Let's say that, in any equation, repeated solutions are given by Sage only once.

Then, how can I, in our example, obtain the following complete solution?

[x == -1, x == -1, x == 1, x == 1, x == 1]

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In order to cover also the multiplicities of the solutions of an algebraic equation, one has the following chances:

$(1)$ Use solve with the option multiplicities=True:

var('x');
f = x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1
solve(f==0, x, multiplicities=True)


This gives:

([x == -1, x == 1], [2, 3])


indicating that the solutions $x=-1$ and respectively $x=1$ have multiplicities $2$, and respectively $3$. To obtain this information, just ask for the doc string of the used function solve:

sage: ?solve

Signature:      solve(f, *args, **kwds)
Docstring:
Algebraically solve an equation or system of equations (over the
complex numbers) for given variables. Inequalities and systems of
inequalities are also supported.

INPUT:

* "f" - equation or system of equations (given by a list or tuple)

* "*args" - variables to solve for.

* "solution_dict" - bool (default: False); if True or non-zero,
return a list of dictionaries containing the solutions. If there
are no solutions, return an empty list (rather than a list
containing an empty dictionary). Likewise, if there's only a
single solution, return a list containing one dictionary with
that solution.

There are a few optional keywords if you are trying to solve a
single equation.  They may only be used in that context.

* "multiplicities" - bool (default: False); if True, return
corresponding multiplicities.  This keyword is incompatible with
"to_poly_solve=True" and does not make any sense when solving
inequalities.


and many other lines.

$(2)$ Use an algebraic object, thus making the used polynomial element in a polynomial ring, then ask for the roots of the polynomial.

R.<x> = PolynomialRing(QQ)
f = x^5 - x^4 - 2*x^3 + 2*x^2 + x - 1
f.roots()


This gives:

[(-1, 2), (1, 3)]


a list of tuples of the shape ( solution, its multiplicity ). So the solution $x=-1$ has multiplicity $2$, and the solution $x=1$ has multiplicity $3$.

This time, if the multiplicities are not wanted, we can ask for this other version of the result...

sage: f.roots(multiplicities=False)
[-1, 1]

more

What about this system of equations, which has a repeated root and and we want to know what it is: var('a,b,c,d,e,f,g,x') sols = solve([(a - g) + 2, - (ag - b) - 1, - (bg - c), - (cg - d) + 10, - (dg - e) - 22, - fg - 2, - (eg - f) + 14], a,b,c,d,e,f,g, solution_dict=true) [{g: sol[g]} for sol in sols]