[tex]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)}[/tex]
Answer:
[tex](a+b)(a^2 -ab + b^2)[/tex]
Step-by-step explanation:
Since, we know that,
[tex](a+b)^2 = a^2 + 2ab + b^2-----(1)[/tex]
Also,
[tex](a+b)^3=a^3 + 3a^2b + 3ab^2 + b^3---(2)[/tex]
Now,
[tex](a+b)^3 = (a+b)(a+b)^2[/tex]
From equation (1) and (2),
[tex]a^3 + 3a^2b + 3ab^2 + b^3 = (a+b)(a^2 + 2ab + b^2)[/tex]
[tex]a^3+b^3 = (a+b)(a^2 + 2ab + b^2)-3a^2b - 3ab^2[/tex]
[tex]a^3 + b^3 = (a+b)(a^2 + 2ab + b^2)-3ab(a +b)[/tex]
[tex]a^3 + b^3 = (a+b)(a^2 + 2ab + b^2-3ab)[/tex]
[tex]a^3 + b^3 = (a+b)(a^2 -ab + b^2)[/tex]
Hence, THIRD option is correct.
