$$-80x^3+32x^2+5x-2 = 0$$

$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 4 } & x_2 = -\dfrac{ 1 }{ 4 } & x_3 = \dfrac{ 2 }{ 5 } \end{matrix} $$

$ \color{blue}{ -80x^{3}+32x^{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 factors of the leading coefficient ( -80 ) are 1 2 4 5 8 10 16 20 40 80 .The factors of the constant term (-2) are 1 2 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 5 } , ~ \pm \frac{ 1 }{ 8 } , ~ \pm \frac{ 1 }{ 10 } , ~ \pm \frac{ 1 }{ 16 } , ~ \pm \frac{ 1 }{ 20 } , ~ \pm \frac{ 1 }{ 40 } , ~ \pm \frac{ 1 }{ 80 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 4 } , ~ \pm \frac{ 2 }{ 5 } , ~ \pm \frac{ 2 }{ 8 } , ~ \pm \frac{ 2 }{ 10 } , ~ \pm \frac{ 2 }{ 16 } , ~ \pm \frac{ 2 }{ 20 } , ~ \pm \frac{ 2 }{ 40 } , ~ \pm \frac{ 2 }{ 80 } ~ $$

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(\frac{ 1 }{ 4 }) = 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}{ 4 x - 1 } $

$$ \frac{ -80x^{3}+32x^{2}+5x-2 }{ \color{blue}{ 4x - 1 } } = -20x^{2}+3x+2 $$

Polynomial $ -20x^{2}+3x+2 $ can be used to find the remaining roots.

$ \color{blue}{ -20x^{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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