Tham khảo nha \(\)

1. Rút gọn:

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

= \(x^3+3x^2+9x-3x^2-9x-27+54+x^3\)

= \(2x^3+27\)

b/ \(\left(3x+y\right)\left(9x^2-3xy+y^2\right)-\left(3x-y\right)\left(9x^2+3xy+y^2\right)\)

\(=27x^3-9x^2y+3xy^2+9x^2y-3xy^2+y^3-27x^3+9x^2y+3xy^2-9x^2y-3xy^2-y^3\)

\(=\left(27x^3-y^3\right)-\left(27x^3+y^3\right)\)

\(=27x^3-y^3-27x^3-y^3=-2y^3\)

2.Chứng minh rằng:

a/ \(a^3+b^3=\left(a+b\right)^3-3ab\left(a+b\right)\)

Xét VP có:

\(=a^3+3a^2b+3ab^2+b^3-3a^2b-3ab^2\)

\(=a^3+b^3\)

=> VT=VP

=> \(a^3+b^3=\left(a+b\right)^3-3ab\left(a+b\right)\)

b/ \(a^3-b^3=\left(a-b\right)^3+3ab\left(a-b\right)\)

Xét VP có:

\(=a^3-3a^2b+3ab^2-b^3+3a^2b-3ab^2\)

\(=a^3-b^3\)

=> VT=VP

=> \(a^3-b^3=\left(a-b\right)^3+3ab\left(a-b\right)\)