rút gọn: (x ² - 5x + 1) ² + 2(5x-1)(x ² - 5x + 1) + (5x-1) ² CMR: 1000² + 1003 ² + 1005 ² + 1006 ²= 1001 ² + 1002 ² + 1004 ² + 1007 ²

Đáp án:

$\begin{array}{l}
a){\left( {{x^2} - 5x + 1} \right)^2} + 2.\left( {5x - 1} \right)\left( {{x^2} - 5x + 1} \right) + {\left( {5x - 1} \right)^2}\\
 = {\left( {{x^2} - 5x + 1 + 5x - 1} \right)^2}\\
 = {\left( {{x^2}} \right)^2}\\
 = {x^4}\\
b)Giả\,sử:\\
{1000^2} + {1003^2} + {1005^2} + {1006^2}\\
 = {1001^2} + {1002^2} + {1004^2} + {1007^2}\\
 \Leftrightarrow {1001^2} - {1000^2} + {1007^2} - {1006^2}\\
 = {1003^2} - {1002^2} + {1005^2} - {1004^2}\\
 \Leftrightarrow \left( {1001 - 1000} \right)\left( {1001 + 1000} \right) + \left( {1007 - 1006} \right)\left( {1007 + 1006} \right)\\
 = \left( {1003 - 1002} \right)\left( {1003 + 1002} \right) + \left( {1005 - 1004} \right)\left( {1005 + 1004} \right)\\
 \Leftrightarrow 2001 + 2013 = 2005 + 2009\\
 \Leftrightarrow 4014 = 4014\left( {tmdk} \right)\\
Vay\,{1000^2} + {1003^2} + {1005^2} + {1006^2}\\
 = {1001^2} + {1002^2} + {1004^2} + {1007^2}
\end{array}$