Question

The factorized form of a³ + b³ + 3ab – 1 is ____________.

Answer 1

$$\begin{gathered} a^3+b^3+3ab-1 \\ =a^3+b^3-1+3ab \\ =a^3+b^3+{\left(-1\right)}^3-3ab\left(-1\right) \\ =\left(a+b-1\right)\left(a^2+b^2+{\left(-1\right)}^2-ab-b\left(-1\right)-a\left(-1\right)\right)\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)\right) \\ =\left(a+b-1\right)\left(a^2+b^2+1-ab+b+a\right) \end{gathered}$$

Hence, the factorized form of a³ + b³ + 3ab – 1 is $\underline{\left(a+b-1\right)\left(a^2+b^2+1-ab+b+a\right)}.$