$ \color{blue}{ -10000x^{3}+28334x^{2}-18335x+3334 } $ 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 ( -10000 ) are 1 2 4 5 8 10 16 20 25 40 50 80 100 125 200 250 400 500 625 1000 1250 2000 2500 5000 10000 .The factors of the constant term (3334) are 1 2 1667 3334 . 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 }{ 25 } , ~ \pm \frac{ 1 }{ 40 } , ~ \pm \frac{ 1 }{ 50 } , ~ \pm \frac{ 1 }{ 80 } , ~ \pm \frac{ 1 }{ 100 } , ~ \pm \frac{ 1 }{ 125 } , ~ \pm \frac{ 1 }{ 200 } , ~ \pm \frac{ 1 }{ 250 } , ~ \pm \frac{ 1 }{ 400 } , ~ \pm \frac{ 1 }{ 500 } , ~ \pm \frac{ 1 }{ 625 } , ~ \pm \frac{ 1 }{ 1000 } , ~ \pm \frac{ 1 }{ 1250 } , ~ \pm \frac{ 1 }{ 2000 } , ~ \pm \frac{ 1 }{ 2500 } , ~ \pm \frac{ 1 }{ 5000 } , ~ \pm \frac{ 1 }{ 10000 } , ~ \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 }{ 25 } , ~ \pm \frac{ 2 }{ 40 } , ~ \pm \frac{ 2 }{ 50 } , ~ \pm \frac{ 2 }{ 80 } , ~ \pm \frac{ 2 }{ 100 } , ~ \pm \frac{ 2 }{ 125 } , ~ \pm \frac{ 2 }{ 200 } , ~ \pm \frac{ 2 }{ 250 } , ~ \pm \frac{ 2 }{ 400 } , ~ \pm \frac{ 2 }{ 500 } , ~ \pm \frac{ 2 }{ 625 } , ~ \pm \frac{ 2 }{ 1000 } , ~ \pm \frac{ 2 }{ 1250 } , ~ \pm \frac{ 2 }{ 2000 } , ~ \pm \frac{ 2 }{ 2500 } , ~ \pm \frac{ 2 }{ 5000 } , ~ \pm \frac{ 2 }{ 10000 } , ~ \pm \frac{ 1667 }{ 1 } , ~ \pm \frac{ 1667 }{ 2 } , ~ \pm \frac{ 1667 }{ 4 } , ~ \pm \frac{ 1667 }{ 5 } , ~ \pm \frac{ 1667 }{ 8 } , ~ \pm \frac{ 1667 }{ 10 } , ~ \pm \frac{ 1667 }{ 16 } , ~ \pm \frac{ 1667 }{ 20 } , ~ \pm \frac{ 1667 }{ 25 } , ~ \pm \frac{ 1667 }{ 40 } , ~ \pm \frac{ 1667 }{ 50 } , ~ \pm \frac{ 1667 }{ 80 } , ~ \pm \frac{ 1667 }{ 100 } , ~ \pm \frac{ 1667 }{ 125 } , ~ \pm \frac{ 1667 }{ 200 } , ~ \pm \frac{ 1667 }{ 250 } , ~ \pm \frac{ 1667 }{ 400 } , ~ \pm \frac{ 1667 }{ 500 } , ~ \pm \frac{ 1667 }{ 625 } , ~ \pm \frac{ 1667 }{ 1000 } , ~ \pm \frac{ 1667 }{ 1250 } , ~ \pm \frac{ 1667 }{ 2000 } , ~ \pm \frac{ 1667 }{ 2500 } , ~ \pm \frac{ 1667 }{ 5000 } , ~ \pm \frac{ 1667 }{ 10000 } , ~ \pm \frac{ 3334 }{ 1 } , ~ \pm \frac{ 3334 }{ 2 } , ~ \pm \frac{ 3334 }{ 4 } , ~ \pm \frac{ 3334 }{ 5 } , ~ \pm \frac{ 3334 }{ 8 } , ~ \pm \frac{ 3334 }{ 10 } , ~ \pm \frac{ 3334 }{ 16 } , ~ \pm \frac{ 3334 }{ 20 } , ~ \pm \frac{ 3334 }{ 25 } , ~ \pm \frac{ 3334 }{ 40 } , ~ \pm \frac{ 3334 }{ 50 } , ~ \pm \frac{ 3334 }{ 80 } , ~ \pm \frac{ 3334 }{ 100 } , ~ \pm \frac{ 3334 }{ 125 } , ~ \pm \frac{ 3334 }{ 200 } , ~ \pm \frac{ 3334 }{ 250 } , ~ \pm \frac{ 3334 }{ 400 } , ~ \pm \frac{ 3334 }{ 500 } , ~ \pm \frac{ 3334 }{ 625 } , ~ \pm \frac{ 3334 }{ 1000 } , ~ \pm \frac{ 3334 }{ 1250 } , ~ \pm \frac{ 3334 }{ 2000 } , ~ \pm \frac{ 3334 }{ 2500 } , ~ \pm \frac{ 3334 }{ 5000 } , ~ \pm \frac{ 3334 }{ 10000 } ~ $$

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 }{ 2 }) = 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}{ 2 x - 1 } $

$$ \frac{ -10000x^{3}+28334x^{2}-18335x+3334 }{ \color{blue}{ 2x - 1 } } = -5000x^{2}+11667x-3334 $$

Polynomial $ -5000x^{2}+11667x-3334 $ can be used to find the remaining roots.

$ \color{blue}{ -5000x^{2}+11667x-3334 } $ 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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