Question

Factorize the following: $8a^3+b^3+12a^{2}b+6a{b}^2$

Answer 1

The expression $8a^3+b^3+12a^{2}b+6a{b}^2$ can be written as ${\left(2a\right)}^3+b^3+3{\left(2a\right)}^{2}b+3\left(2a\right){\left(b\right)}^2$ ….[1]

We know the Algebraic Identity ${\left(a+b\right)}^3=a^3+b^3+3ab\left(a+b\right)$

$=a^3+b^3+3a^{2}b+3a{b}^2$ …[2]

By comparing expressions [1] and [2], we can say,

$8a^3+b^3+12a^{2}b+6a{b}^2$ $=$${\left(2a\right)}^3+b^3+3{\left(2a\right)}^{2}b+3\left(2a\right){\left(b\right)}^2$

$={\left(2a+b\right)}^3$

Hence, $\mathbf{8}{\mathit{a}}^{\mathbf{3}}\mathbf{+}{\mathit{b}}^{\mathbf{3}}\mathbf{+}\mathbf{12}{\mathit{a}}^{\mathbf{2}}\mathit{b}\mathbf{+}\mathbf{6}\mathit{a}{\mathit{b}}^{\mathbf{2}}\mathbf{=}\left(2a+b\right)\mathbf{(}\mathbf{2}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}$