A real root of an equation 
is a root belonging to the real numbers. A polynomial
with real coefficients
can also have nonreal complex roots,
which occur in complex conjugate pairs.
A nonzero polynomial whose complex roots are all real is called a real-rooted polynomial.
The number of real roots of a nonzero real polynomial, counted with multiplicity, has the
same parity as its polynomial
degree. Indeed, the fundamental theorem
of algebra gives 
roots for polynomial degree 
, counted with multiplicity,
and the nonreal roots occur in complex
conjugate pairs. In particular, every real polynomial
of odd polynomial degree has at least one real
root (Trott 2006). A real polynomial of even polynomial degree need not have any real roots,
as shown by 
.
For a real polynomial without repeated roots, the Sturm theorem counts its real roots in an interval whose endpoints are not roots. For polynomials with parameters, quantifier elimination can express conditions on the coefficients for a specified number of distinct real roots (Yang 1999, Trott 2006).
