Question
$$\frac{x^3}{(x-1)(x-2)} = 0$$
Answer
$$ \begin{matrix}x_1 = 0 & x_2 = 2 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x^3}{(x-1)(x-2)} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x-1)(x-2) }. \\[1 em](x-1)(x-2)\frac{x^3}{(x-1)(x-2)} &= (x-1)(x-2)\cdot0&& \text{cancel out the denominators} \\[1 em]x^5-4x^4+4x^3 &= 0&& \\[1 em] \end{aligned} $$
In order to solve $ \color{blue}{ x^{5}-4x^{4}+4x^{3} = 0 } $, first we need to factor our $ x^3 $.
$$ x^{5}-4x^{4}+4x^{3} = x^3 \left( x^{2}-4x+4 \right) $$$ x = 0 $ is a root of multiplicity $ 3 $.
The remaining roots can be found by solving equation $ x^{2}-4x+4 = 0$.
$ x^{2}-4x+4 = 0 $ is a quadratic equation.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this equation.
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