Rút gọn các biểu thức sau:

$a) (x+3\left)\right(x^2-3x+9)-(54+x^3)$

$b) (2x+y\left)\right(4x^2-2xy+y^2)-(2x-y\left)\right(4x^2+2xy+y^2)$

Giải:

$$\begin{gathered} a) (x+3\left)\right(x^2-3x+9)-(54+x^3) \\ = (x+3\left)\right(x^2-3x+3^2)-(54+x^3) \\ = x^3+3^3-(54+x^3) \\ = x^3+27- 54-x^3 \\ =-27 \end{gathered}$$

$$\begin{gathered} b) (2x+y\left)\right(4x^2-2xy+y^2)-(2x-y\left)\right(4x^2+2xy+y^2) \\ = (2x+y\left)\right[{\left(2x\right)}^2-2.x.y+y^2]-(2x-y){\left(2x\right)}^2+2.x.y+y^2] \\ =[{\left(2x\right)}^3+y^3]-[{\left(2x\right)}^3-y^3] \\ = {\left(2x\right)}^3+y^3-{\left(2x\right)}^3+y^3 \\ =2y^3 \end{gathered}$$