Thu gọn các biểu thức sau:
a) \(20{x^2} - \left( {5x - 4} \right)\left( {4 + 5x} \right)\);
b) \({\left( {x - y} \right)^2} - x\left( {x + 2y} \right)\);
c) \({\left( {x + 3} \right)^3} - {\left( {x - 3} \right)^3}\);
d) \(x\left( {x - 1} \right)\left( {x + 1} \right) - \left( {x - 3} \right)\left( {{x^2} + 3x + 9} \right)\).
Sử dụng kiến thức về hằng đẳng thức để thu gọn biểu thức:
a) \(\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\)
b) \({\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\)
c) \({a^3} - 3{a^2}b + 3a{b^2} - {b^3} = {\left( {a - b} \right)^3}\); \({a^3} + 3{a^2}b + 3a{b^2} + {b^3} = {\left( {a + b} \right)^3}\)
d) \(\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\); \({a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\)
a) \(20{x^2} - \left( {5x - 4} \right)\left( {4 + 5x} \right) = 20{x^2} - \left[ {{{\left( {5x} \right)}^2} - {4^2}} \right] = 20{x^2} - 25{x^2} + 16 = - 5{x^2} + 16\);
b) \({\left( {x - y} \right)^2} - x\left( {x + 2y} \right) = {x^2} - 2xy + {y^2} - {x^2} - 2xy\)
\( = \left( {{x^2} - {x^2}} \right) - \left( {2xy + 2xy} \right) + {y^2} = - 4xy + {y^2}\)
c) \({\left( {x + 3} \right)^3} - {\left( {x - 3} \right)^3} = {x^3} + 3.{x^2}.3 + 3.x{.3^2} + {3^3} - \left( {{x^3} - 3.{x^2}.3 + 3.x{{.3}^2} - {3^3}} \right)\)
\( = {x^3} + 9{x^2} + 27x + 27 - {x^3} + 9{x^2} - 27x + 27\)
\( = \left( {{x^3} - {x^3}} \right) + \left( {9{x^2} + 9{x^2}} \right) + \left( {27x - 27x} \right) + \left( {27 + 27} \right) = 18{x^2} + 54\)
d) \(x\left( {x - 1} \right)\left( {x + 1} \right) - \left( {x - 3} \right)\left( {{x^2} + 3x + 9} \right) = x\left( {{x^2} - 1} \right) - \left( {{x^3} - {3^3}} \right) = {x^3} - x - {x^3} + 27\)
\( = \left( {{x^3} - {x^3}} \right) - x + 27 = - x + 27\)