1)
$\begin{array}{l} P = 1 + x + {x^2} + ... + {x^9} + {x^{10}}\\ \Rightarrow x.P = x + {x^2} + {x^3} + ... + {x^{11}}\\ \Rightarrow xP - P = {x^{11}} - 1 \end{array}$
2)
$\begin{array}{l} P = {x^{10}} - 10{x^9} + 10{x^8} - 10{x^7} + ... - 10x + 9\\ P = {x^{10}} - \left( {9 + 1} \right){x^9} + \left( {9 + 1} \right){x^8} - \left( {9 + 1} \right){x^7} + ... + {x^2}\left( {9 + 1} \right) - \left( {9 + 1} \right)x + 9\left( {x = 9 \Rightarrow 10 = x + 1} \right)\\ P = {x^9}\left( {x - 9} \right) - {x^8}\left( {x - 9} \right) + {x^7}\left( {x - 9} \right) - ... + x\left( {x - 9} \right) - x + 10\\ P = {9^9}\left( {9 - 9} \right) - {9^8}\left( {9 - 9} \right) + {9^7}\left( {9 - 9} \right) - ... + 9\left( {9 - 9} \right) - 9 + 10 = 1 \end{array}$
3)
$\begin{array}{l} S = 1 + 2 + {2^2} + {2^3} + ... + {2^{12}} + {2^{13}} + {2^{14}}\\ S = \left( {1 + 2 + {2^2} + {2^3} + {2^4}} \right) + \left( {{2^5} + {2^6} + {2^7} + {2^8} + {2^9}} \right) + \left( {{2^{10}} + {2^{11}} + {2^{12}} + {2^{13}} + {2^{14}}} \right)\\ S = 31 + {2^5}.31 + {2^{10}}.31 \vdots 31 \end{array}$
