How do you factor completely 125x^3+1 ?

Mar 1, 2017

$\left(5 x + 1\right) \left(25 {x}^{2} - 5 x + 1\right)$

Explanation:

$125 {x}^{3} + 1 \text{ is a " color(blue)"sum of cubes}$ and is factorised, in general, as shown below.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{a}^{3} + {b}^{3} = \left(a + b\right) \left({a}^{2} - a b + {b}^{2}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$125 {x}^{3} = {\left(5 x\right)}^{3} \text{ and } 1 = {\left(1\right)}^{3}$

$\text{here "a=5x" and } b = 1$

$\Rightarrow 125 {x}^{3} + 1 = \left(5 x + 1\right) \left(25 {x}^{2} - 5 x + 1\right)$