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If

$$ \tag{* }
f _ {0} ( x) \dots f _ {s} ( x)
$$

is a Sturm series on the interval  $  [ a, b] $, 
$  a < b $, 
and  $  w( x) $
is the number of variations of sign in the series (*) at a point  $  x \in [ a, b] $(
vanishing terms are not taken into consideration), then the number of distinct roots of the function  $  f _ {0} $
on the interval  $  [ a, b] $
is equal to the difference  $  w( a)- w( b) $.

A Sturm series (or Sturm sequence) is a sequence of real-valued continuous functions (*) on  $  [ a, b] $
having a finite number of roots on this interval, and such that

1)  $  f _ {0} ( a) f _ {0} ( b) \neq 0 $;

2)  $  f _ {s} ( x) \neq 0 $
on  $  [ a, b] $;

3) from  $  f _ {k} ( c) = 0 $
for some  $  k $
$  ( 0 < k < s) $
and given  $  c $
in  $  [ a, b] $
it follows that  $  f _ {k-} 1 ( c) f _ {k+} 1 ( c) < 0 $;

4) from  $  f _ {0} ( c) = 0 $
for a given  $  c $
$  ( a < c < b) $
it follows that for sufficiently small  $  \epsilon > 0 $,

$$ 
f _ {0} ( x) f _ {1} ( c)  <  0 \  ( c- \epsilon < x < c);
$$

$$ 
f _ {0} ( x) f _ {1} ( c)  >  0 \  ( c < x < c + \epsilon ).
$$

This theorem was proved by J.Ch. Sturm [[#References|[1]]], who also proposed the following method of constructing a Sturm series for a polynomial  $  f ( x) $
with real coefficients and without multiple roots:  $  f _ {0} ( x) = f ( x) $, 
$  f _ {1} ( x) = f ^ { \prime } ( x) $, 
and, if the polynomials  $  f _ {0} ( x) \dots f _ {k} ( x) $
are already constructed, then as  $  f _ {k+} 1 ( x) $
one should take minus the remainder occurring in the process of dividing  $  f _ {k-} 1 ( x) $
by  $  f _ {k} ( x) $. 
Here,  $  f _ {s} ( x) $
will be a non-zero constant.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.Ch. Sturm,   ''Bull. de Férussac'' , '''11'''  (1829)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,   "Higher algebra" , MIR  (1972)  (Translated from Russian)</TD></TR></table>

====Comments====
The coefficients of the polynomials in the Sturm series must belong to a real-closed field. The algorithm to determine a Sturm series for a polynomial  $  f _ {0} ( x) $
can be described as follows:

$$ 
f _ {0} ( x)  =  f ( x),\  f _ {1} ( x)  =  f ^ { \prime } ( x) ,
$$

$$ 
f _ {0} ( x)  =  q _ {1} ( x) f _ {1} ( x) - f _ {2} ( x)
,\   \mathop{\rm deg}  f _ {2} ( x) <  \mathop{\rm deg}  f _ {1} ( x) ,
$$

$$ 
\dots \dots \dots \dots
$$

$$ 
f _ {k-} 1 ( x)  =  q _ {k} ( x) f _ {k} ( x) - f _ {k+} 1 ( x)
,\   \mathop{\rm deg}  f _ {k+} 1 ( x) <  \mathop{\rm deg}  f _ {k} ( x),
$$

$$ 
\dots \dots \dots \dots
$$

$$ 
f _ {s-} 1 ( x)  =  q _ {s} ( x) f _ {s} ( x) ,
$$

so  $  f _ {s} ( x) $
is a non-zero constant.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,   "Basic algebra" , '''I''' , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.E.J. Dickson,   "New first course in the theory of equations" , Wiley  (1939)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1''' , Springer  (1967)  (Translated from German)</TD></TR></table>