Using binomial theorem, expand each of the following: `(3x^(2)-2ax+3a^(2))^(3)`

Answer

`27x^(6)-54ax^(5)+ 117a^(2)x^(4)-116a^(3)x^(3) + 117a^(4)x^(2)-54a^(5)x+27a^(6)`

Explanation

`(3x^(2) - 2ax + 3a^(2))^(3) = [(3x^(2) - 2ax)+3a^(2)]^(3)`
`=.^(3) C _(0) (3x^(2) - 2ax)^(3) + .^(3)C_(1) (3x^(2) - 2ax)^(2)(3a^(2))+.^(3)C_(2) (3x^(2)-2ax) (3a^(2))^(2) + .^(3)C_(3)(3a^(2))^(3)`
`=[.^(3)C_(0)(3x^(2))^(3) -.^(3)C_(1)(3x^(2))^(2) xx2ax+.^(3)C_(2)(3x^(2)) xx (2ax)^(2) - .^(3)C_(3)(2ax)^(3)]`
`+(9x^(4) + 4a^(2)x^(2) - 12ax^(3)) (3a^(2)) + (27a^(4)) (3x^(2) - 2ax) + 27a^(6)`
`=27x^(6) - 54ax^(5) + 36a^(2)x^(4) - 8a^(3)x^(3) + 81a^(2)x^(4) +36a^(4)x^(2) - 108a^(3) x^(3) + 81a^(4)x^(2)-54a^(5) x + 27a^(6)`
`=27x^(6) - 54ax^(5) +117a^(2)x^(2) -116a^(3)x^(3) + 117a^(4)x^(2) - 54a^(5)x = 27a^(6).`