$$-x^5+4x^4-5x^3+2x^2 = 0$$

$$ \begin{matrix}x_1 = 0 & x_2 = 1 & x_3 = 2 \end{matrix} $$

In order to solve $ \color{blue}{ -x^{5}+4x^{4}-5x^{3}+2x^{2} = 0 } $, first we need to factor our $ x^2 $.

$$ -x^{5}+4x^{4}-5x^{3}+2x^{2} = x^2 \left( -x^{3}+4x^{2}-5x+2 \right) $$

$ x = 0 $ is a root of multiplicity $ 2 $.

The remaining roots can be found by solving equation $ -x^{3}+4x^{2}-5x+2 = 0$.

$ \color{blue}{ -x^{3}+4x^{2}-5x+2 } $ is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $ \dfrac{p}{q} $, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factor of the leading coefficient ( -1 ) is 1 .The factors of the constant term (2) are 1 2 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } ~ $$

Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial $ P(x) $, we obtain $ P(1) = 0 $.

To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $ \color{blue}{q x - p} $. In this example:

Divide $ P(x) $ with $ \color{blue}{x - 1} $

$$ \frac{ -x^{3}+4x^{2}-5x+2 }{ \color{blue}{ x - 1 } } = -x^{2}+3x-2 $$

Polynomial $ -x^{2}+3x-2 $ can be used to find the remaining roots.

$ \color{blue}{ -x^{2}+3x-2 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.

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