By giving explicit upper bounds, the author shows that the Bernstein constants \[ B_{\lambda,p} := \lim_{n\to\infty} n^{\lambda+1/p} \inf_{c_k} \Biggl\| | x| ^\lambda - \sum^n_{k=0} c_k x^k\Biggl\|_{L_p[-1,1]} \] are finite for all \(\lambda > 0\) and \(p\in (1/3,\infty)\). For \(p = 1\), the upper bounds turn out to be sharp. The main result follows from asymptotic relations for the error of approximation of \(| x| ^\lambda\) in \(L_p [-1,1]\) by polynomials of even degree given as the sum of the Lagrange interpolation polynomial to \(| x| ^\lambda\) at the Chebyshev nodes of 1st and 2nd kind and certain scalar multiples of the Chebyshev polynomials of 1st and 2nd kind (with scalars depending on \(n\) and \(\lambda\)). Such asymptotic relations are also given for the case \(p=\infty\), that is, for uniform approximation.