This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# CountRoots

 CountRootsgives the number of real roots of the polynomial poly in x. CountRoots gives the number of roots between a and b.
• CountRoots enumerates roots with multiplicities counted.
• Roots that lie exactly at or are counted.
• The limits a and b can be complex, in which case the range is taken to be a closed rectangle in the complex plane.
Count the number of polynomial roots between 0 and 10:
Count roots of a polynomial in a closed rectangle:
Count the number of polynomial roots between 0 and 10:
 Out[1]=
Count roots of a polynomial in a closed rectangle:
 Out[2]=
 Scope   (8)
Count roots in a real interval:
Find the number of the real roots:
Count roots in a closed rectangle:
Count roots in a vertical line segment:
Count roots in a horizontal line segment:
Multiple roots are counted with their multiplicities:
Roots at the endpoints of the interval are included:
Roots on the boundary of the rectangle are included:
Count the real roots of a high-degree polynomial:
Find the number of real roots of an algebraic function involving high-degree radicals:
Count the real roots of a function involving irrational real powers:
Count the real roots of an exp-log function:
Count roots of an elementary function in a bounded interval:
This shows the plot of the function:
Count roots of a holomorphic elementary function in a closed rectangle:
 Applications   (1)
The number of 17 roots of unity in the closed unit square in the first quadrant:
Roots on the boundary are counted:
The number of complex roots of a polynomial is equal to its degree:
This gives a bound on absolute values of roots of a polynomial:
The polynomial indeed has 10 roots within the Cauchy bounded region:
The number of real roots of a polynomial with nonzero terms is at most :
This polynomial has the maximal possible number of real roots:
Use Reduce to find polynomial roots:
Count roots:
Find roots:
Use RootIntervals to find isolating intervals for polynomial roots:
Count the real roots:
Isolate the real roots:
Use NumberFieldSignature to count the real roots and the pairs of complex roots:
Roots at the endpoints of the interval are included:
New in 6