3. Let \(T_{0}=1, T_{1}=x, T_{k+1}=2 x T_{k}-T_{k-1}\), where \(k \in N\). Write in standard form \(T_{5}\) and \(T_{0}+T_{1}+T_{2}+T_{3}\).

The polynomials mentioned above are known as "Chebyshev polynomials" (1821 - 1894). The discovery of these polynomials and a series of their remarkable properties was prompted by the great Russian mathematician's study of a linkage mechanism called the "Watt's parallelogram" (see, for example, the journal "Kvant" for 1971, issue 5, article by N. N. Kolesnikov "Pafnuty Lvovich Chebyshev").
3. Solution. \(T_{2}=2 x T_{1}-T_{0}=2 x \cdot x-1=2 x^{2}-1\),

\[
T_{3}=4 x^{3}-3 x, \quad T_{4}=8 x^{4}-8 x^{2}+1, \quad T_{5}=16 x^{5}-20 x^{3}+5 x
\]

\[
\begin{aligned}
& T_{0}+T_{1}+T_{2}+T_{3}=1+x+2 x^{2}-1+4 x^{3}- \\
& -3 x=4 x^{3}+2 x^{2}-2 x
\end{aligned}
\]