Question

Which of the following expressions is equivalent to a 3 + b 3? (a - b )(a 2 + ab + b 2 ) a 3 + a 2b + ab 2 - a 2b - ab 2 - b 3 (a + b )(a 2 - ab + b 2 ) (a + b )(a 3 + b 3 )

Answer 1

`a^3+b^3=a^3+a^2b-a^2b+ab^2-ab^2+b^3\\\\=(a^3+a^2b)-(a^2b+ab^2)+(ab^2+b^3)\\\\=a^2(a+b)-ab(a+b)+b^2(a+b)\\\\=(a+b)(a^2-ab+b^2)\\\\\\\boxed{a^3+b^3=(a+b)(a^2-ab+b^2)}`

Answer 2

`(a+b)(a^2 -ab + b^2)`

Explanation:

Since, we know that,

`(a+b)^2 = a^2 + 2ab + b^2-----(1)`

Also,

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

Now,

`(a+b)^3 = (a+b)(a+b)^2`

From equation (1) and (2),

`a^3 + 3a^2b + 3ab^2 + b^3 = (a+b)(a^2 + 2ab + b^2)`

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

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

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

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

Hence, THIRD option is correct.