Đề bài

Rút gọn các biểu thức sau:

✱a) \({\left( {x - 2y} \right)^3}\; + {\left( {x + 2y} \right)^3}\).

꧋b) \({\left( {3x + 2y} \right)^3}\; + {\left( {3x - 2y} \right)^3}\).

Lời giải chi tiết

a) Ta có: \({(x - 2y)^3} + {(x + 2y)^3}\)  

ꦕ\(\begin{array}{l} = \left[ {{x^3} - 3.{x^2}.2y + 3.x.{{(2y)}^2} - {{(2y)}^3}} \right] + \left[ {{x^3} + 3.{x^2}.2y + 3.x.{{(2y)}^2} + {{(2y)}^3}} \right]\\ = {x^3} - 6{x^2}y + 12x{y^2} - 8{y^3} + {x^3} + 6{x^2}y + 12x{y^2} + 8{y^3}\\ = \left( {{x^3} + {x^3}} \right) + \left( { - 6{x^2}y + 6{x^2}y} \right) + \left( {12x{y^2} + 12x{y^2}} \right) + \left( { - 8{y^3} + 8{y^3}} \right)\\ = 2{x^3} + 24x{y^2}\end{array}\)

b) Ta có: \({(3x + 2y)^3} + {(3x - 2y)^3}\)

🍸\( = \left[ {{{(3x)}^3} + 3.{{(3x)}^2}.2y + 3.3x.{{(2y)}^2} + {{(2y)}^3}} \right] + \left[ {{{(3x)}^3} - 3.{{(3x)}^2}.2y + 3.3x.{{(2y)}^2} - {{(2y)}^3}} \right]\)\( = 27{x^3} + 54{x^2}y + 36x{y^2} + 8{y^3} + 27{x^3} - 54{x^2}y + 36x{y^2} - 8{y^3}\)\( = \left( {27{x^3} + 27{x^3}} \right) + \left( {54{x^2}y - 54{x^2}y} \right) + \left( {36x{y^2} + 36x{y^2}} \right) + \left( {8{y^3} - 8{y^3}} \right)\)\( = 54{x^3} + 72x{y^2}.\)