BUILT-IN MATHEMATICA SYMBOL

CountRoots

gives the number of real roots of the polynomial poly in x.

CountRoots

gives the number of roots between a and b.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (1)

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:

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Out[1]=

Count roots of a polynomial in a closed rectangle:

In[2]:=

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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:

Generalizations & Extensions (6)

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:

Properties & Relations (5)

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:

Possible Issues (1)

Roots at the endpoints of the interval are included: