Editing Sturm's theorem (section)

==Example==
Suppose we wish to find the number of roots in some range for the polynomial <math>p(x)=x^4+x^3-x-1</math>. So

:<math>\begin{align} p_0(x) &=p(x)=x^4+x^3-x-1 \\
p_1(x)&=p'(x)=4x^3+3x^2-1
\end{align}</math>

The remainder of the [[Euclidean division of polynomials|Euclidean division]] of {{math|''p''<sub>0</sub>}} by {{math|''p''<sub>1</sub>}} is <math>-\tfrac{3}{16}x^2-\tfrac{3}{4}x-\tfrac{15}{16};</math> multiplying it by {{math|−1}} we obtain 
:<math>p_2(x)=\tfrac{3}{16}x^2+\tfrac{3}{4}x+\tfrac{15}{16}</math>.  
Next dividing {{math|''p''<sub>1</sub>}} by {{math|''p''<sub>2</sub>}} and multiplying the remainder by {{math|−1}}, we obtain 
:<math>p_3(x)=-32x-64</math>. 
Now dividing {{math|''p''<sub>2</sub>}} by {{math|''p''<sub>3</sub>}} and multiplying the remainder by {{math|−1}}, we obtain 
:<math>p_4(x)=-\tfrac{3}{16}</math>.
As this is a constant, this finishes the computation of the Sturm sequence.

To find the number of real roots of <math>p_0</math> one has to evaluate the sequences of the signs of these polynomials at {{math|−∞}} and {{math|∞}}, which are respectively {{math|(+, −, +, +, −)}} and {{math|(+, +, +, −, −)}}. Thus
:<math>V(-\infty)-V(+\infty) = 3-1=2,</math>
where {{mvar|V}} denotes the number of sign changes in the sequence, which shows that {{mvar|p}} has two real roots.

This can be verified by noting that {{math|''p''(''x'')}} can be factored as {{math|(''x''<sup>2</sup> − 1)(''x''<sup>2</sup> + ''x'' + 1)}}, where the first factor  has the roots {{math|−1}} and {{math|1}}, and second factor has no real roots. This last assertion results from the [[quadratic formula]], and also from Sturm's theorem, which gives the sign sequences {{math|(+, –, –)}} at {{math|−∞}} and {{math|(+, +, –)}} at {{math|+∞}}.