  We investigate an infinite sequence of polynomials of the form:
\[a_0T_{n}(x)+a_{1}T_{n-1}(x)+\cdots+a_{m}T_{n-m}(x)\] where
$(a_0,a_1,\ldots,a_m)$ is a fixed m-tuple of real numbers, $a_0,a_m\ne0$,
$T_i(x)$ are Chebyshev polynomials of the first kind, $n=m,m+1,m+2,\ldots$ Here
we analyse the structure of the set of zeros of such polynomial, depending on
$A$ and its limit points when $n$ tends to infinity. Also the expression of
envelope of the polynomial is given. An application in number theory, more
precise, in the theory of Pisot and Salem numbers is presented.
