Question

Factorize the expression $125x^3+\frac{1}{8}$.

Answer 1

Factorizing the given expression:

The given expression is $125x^3+\frac{1}{8}$.

It can be rewritten as $125x^3+\frac{1}{8}={\left(5x\right)}^3+{\left(\frac{1}{2}\right)}^3$.

We know identity, $a^3+b^3=(a+b)\left(a^2-ab+b^2\right)$.

Comparing expression with identity we get, $a=5x,b=\frac{1}{2}$.

Substituting values in the identity we get:

$$\begin{gathered} {\left(5x\right)}^3+{\left(\frac{1}{2}\right)}^3=\left(5x+\frac{1}{2}\right)\left[{\left(5x\right)}^2-\left(5x\right)\left(\frac{1}{2}\right)+{\left(\frac{1}{2}\right)}^2\right] \\ =\left(5x+\frac{1}{2}\right)\left(25x^2-\frac{5x}{2}+\frac{1}{4}\right) \end{gathered}$$

Hence, the expression, $125x^3+\frac{1}{8}=\left(5x+\frac{1}{2}\right)\left(25x^2-\frac{5x}{2}+\frac{1}{4}\right)$.