Rút gọn các biểu thức:
a) \({(x + 1)^2} - \left( {x + 3} \right)\left( {x - 3} \right) - 10\)
b) \(\left( {x + 5} \right)\left( {{x^2} - 5x + 25} \right) - x{\left( {x - 4} \right)^2} + 16x\)
c) \({\left( {x - 2y} \right)^3} - \left( {x + 2y} \right)\left( {{x^2} - 2xy + 4{y^2}} \right) + 6{x^2}y\)
Sử dụng các hằng đẳng thức đáng nhớ.
a) \({(x + 1)^2} - \left( {x + 3} \right)\left( {x - 3} \right) - 10\)
\(\begin{array}{l} = {\left( {x + 1} \right)^2} - \left( {{x^2} - {3^2}} \right) - 10\\ = {x^2} + 2x + 1 - {x^2} + 9 - 10\\ = \left( {{x^2} - {x^2}} \right) + 2x + \left( {1 + 9 - 10} \right)\\ = 2x\end{array}\)
b) \(\left( {x + 5} \right)\left( {{x^2} - 5x + 25} \right) - x{\left( {x - 4} \right)^2} + 16x\)
\(\begin{array}{l} = {x^3} + {5^3} - x\left( {{x^2} - 8x + 16} \right) + 16x\\ = {x^3} + 125 - {x^3} + 8{x^2} - 16x + 16x\\ = 8{x^2} + 125\end{array}\)
c) \({\left( {x - 2y} \right)^3} - \left( {x + 2y} \right)\left( {{x^2} - 2xy + 4{y^2}} \right) + 6{x^2}y\)
\(\begin{array}{l} = {\left( {x - 2y} \right)^3} - \left( {x + 2y} \right)\left( {{x^2} - 2xy + 4{y^2}} \right) + 6{x^2}y\\ = {x^3} - 6{x^2}y + 12x{y^2} - 8{y^3} - \left( {{x^3} + 8{y^3}} \right) + 6{x^2}y\\ = {x^3} - 6{x^2}y + 12x{y^2} - 8{y^3} - {x^3} - 8{y^3} + 6{x^2}y\\ = \left( {{x^3} - {x^3}} \right) + \left( { - 6{x^2}y + 6{x^2}y} \right) + 12x{y^2} + \left( { - 8{y^3} - 8{y^3}} \right)\\ = 12x{y^2} - 16{y^3}\end{array}\)