Factorize:$ \frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x $

Given:

\( \frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x \)

To do:

We have to factorize the given expression.

Solution:

We know that,

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

$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$

Therefore,

$\frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x =(\frac{2}{3} x)^{3}+(1)^{3}+3 \times (\frac{2}{3} x)^{2} \times 1+3 \times \frac{2}{3} x \times(1)^{2}$

$=(\frac{2}{3} x+1)^{3}$

$=(\frac{2}{3} x+1)(\frac{2}{3} x+1)(\frac{2}{3} x+1)$

Hence, $\frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x = (\frac{2}{3} x+1)(\frac{2}{3} x+1)(\frac{2}{3} x+1)$.