Question

If one factor of $a^3+b^3$ is $(a+b)$, then the other factor is:

• A. $a^3+b^3+ab$
• B. $a-b+ab$
• C. $a^2+b^2-ab$
• D. $a^2+b^2+ab$

Answer 1

The correct option is C $a^2+b^2-ab$

Let's first take $(a+b{)}^3$.

As we know that, $(a+b{)}^3=a^3+b^3+3a{b}^2+3a^{2}b$

$⟹$ $a^3+b^3=(a+b{)}^3-3a{b}^2-3a^{2}b$

$=(a+b{)}^3-3ab(b+a)$

$=(a+b)\left(\right(a+b{)}^2-3ab)$ ................. taking $(a+b)$ common

$=(a+b)(a^2+2ab+b^2-3ab)$ ............. splitting $(a+b{)}^2$

$=(a+b)(a^2-ab+b^2)$.

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

Hence, the another factor is $(a^2+b^2-ab)$.