Example 2. Expand the polynomial $y=x^{3}-2 x^{2}+3 x+5$ in terms of positive integer powers of the binomial $x-2$.
Solution. $y^{\prime}=3 x^{2}-4 x+3, y^{\prime \prime}=6 x-4, y^{\prime \prime \prime}=6, y^{I V}=0$. Next, $y(2)=11, y^{\prime}(2)=7, y^{\prime \prime}(2)=8, y^{\prime \prime \prime}(2)=6$. According to point $2^{\circ}$, we can write $y=11+7(x-2)+4(x-2)^{2}+(x-2)^{3}$. This is what was required.

Answer. $x^{3}-2 x^{2}+3 x+5=11+7(x-2)+4(x-2)^{2}+(x-2)^{3}$.