187k views
0 votes
(a^2+2ab+b^2)^3-27c^6



User Zaffar
by
8.8k points

1 Answer

5 votes

Answer:

The expression you provided is a binomial expression raised to the third power, where the binomial is (a^2 + 2ab + b^2). Using the binomial expansion formula for (a+b)^3, we can simplify the expression as follows:

(a^2 + 2ab + b^2)^3 = [(a+b)^2 + ab]^3 [using the identity (a^2 + 2ab + b^2) = (a+b)^2 + ab]

= [(a+b)^2 + ab][(a+b)^2 + ab][(a+b)^2 + ab]

Expanding each term, we get:

= [(a+b)^4 + 2ab(a+b)^2 + a^2b^2][(a+b)^2 + ab]

= [(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4) + 2ab(a^2 + 2ab + b^2) + a^2b^2][(a+b)^2 + ab]

= [(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4) + 2a^3b + 4a^2b^2 + 2ab^3 + a^2b^2][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 12a^2b^2 + 6ab^3 + b^4 + 3a^2b^2) + (2a^3b + 4a^2b^2 + 2ab^3 + a^2b^2)ab][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 13a^2b^2 + 8ab^3 + b^4) + (2a^4b + 6a^3b^2 + 6a^2b^3 + 2ab^4)][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 13a^2b^2 + 8ab^3 + b^4) + 2ab(a^3 + 3a^2b + 3ab^2 + b^3)][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 13a^2b^2 + 8ab^3 + b^4) + 2ab(a+b)^3][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 13a^2b^2 + 8ab^3 + b^4) + 2ab(a+b)(a+b)^2][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 13a^2b^2 + 8ab^3 + b^4) + 2ab(a+b)(a^2 + 2ab + b^2)][(a+b)^2 + ab]

= [(a^4 + 6a^3b + 13a^2b^2 + 8ab^3 + b^4) + 2a^3b + 4a^2b^2 + 2ab^3 + 2a^2b^2 + 4ab^3 + 2b^4][(a+b)^2 + ab]

User Andrii Kovalenko
by
8.3k points