Consider a polynomial of large degree
n
n
whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly
k
k
real zeros with probability
n
−
b
+
o
(
1
)
n^{-b+o(1)}
as
n
→
∞
n \rightarrow \infty
through integers of the same parity as the fixed integer
k
≥
0
k \ge 0
. In particular, the probability that a random polynomial of large even degree
n
n
has no real zeros is
n
−
b
+
o
(
1
)
n^{-b+o(1)}
. The finite, positive constant
b
b
is characterized via the centered, stationary Gaussian process of correlation function
s
e
c
h
(
t
/
2
)
{\mathrm {sech}} (t/2)
. The value of
b
b
depends neither on
k
k
nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability
n
−
b
+
o
(
1
)
n^{-b+o(1)}
one may specify also the approximate locations of the
k
k
zeros on the real line. The constant
b
b
is replaced by
b
/
2
b/2
in case the i.i.d. coefficients have a nonzero mean.