$ \color{blue}{ -x^{3}+450x^{2}-63125x+2625000 } $ 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 (2625000) are 1 2 3 4 5 6 7 8 10 12 14 15 20 21 24 25 28 30 35 40 42 50 56 60 70 75 84 100 105 120 125 140 150 168 175 200 210 250 280 300 350 375 420 500 525 600 625 700 750 840 875 1000 1050 1250 1400 1500 1750 1875 2100 2500 2625 3000 3125 3500 3750 4200 4375 5000 5250 6250 7000 7500 8750 9375 10500 12500 13125 15000 15625 17500 18750 21000 21875 25000 26250 31250 35000 37500 43750 46875 52500 62500 65625 75000 87500 93750 105000 109375 125000 131250 175000 187500 218750 262500 328125 375000 437500 525000 656250 875000 1312500 2625000 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 7 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 14 }{ 1 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 21 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 25 }{ 1 } , ~ \pm \frac{ 28 }{ 1 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 35 }{ 1 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 42 }{ 1 } , ~ \pm \frac{ 50 }{ 1 } , ~ \pm \frac{ 56 }{ 1 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 70 }{ 1 } , ~ \pm \frac{ 75 }{ 1 } , ~ \pm \frac{ 84 }{ 1 } , ~ \pm \frac{ 100 }{ 1 } , ~ \pm \frac{ 105 }{ 1 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 125 }{ 1 } , ~ \pm \frac{ 140 }{ 1 } , ~ \pm \frac{ 150 }{ 1 } , ~ \pm \frac{ 168 }{ 1 } , ~ \pm \frac{ 175 }{ 1 } , ~ \pm \frac{ 200 }{ 1 } , ~ \pm \frac{ 210 }{ 1 } , ~ \pm \frac{ 250 }{ 1 } , ~ \pm \frac{ 280 }{ 1 } , ~ \pm \frac{ 300 }{ 1 } , ~ \pm \frac{ 350 }{ 1 } , ~ \pm \frac{ 375 }{ 1 } , ~ \pm \frac{ 420 }{ 1 } , ~ \pm \frac{ 500 }{ 1 } , ~ \pm \frac{ 525 }{ 1 } , ~ \pm \frac{ 600 }{ 1 } , ~ \pm \frac{ 625 }{ 1 } , ~ \pm \frac{ 700 }{ 1 } , ~ \pm \frac{ 750 }{ 1 } , ~ \pm \frac{ 840 }{ 1 } , ~ \pm \frac{ 875 }{ 1 } , ~ \pm \frac{ 1000 }{ 1 } , ~ \pm \frac{ 1050 }{ 1 } , ~ \pm \frac{ 1250 }{ 1 } , ~ \pm \frac{ 1400 }{ 1 } , ~ \pm \frac{ 1500 }{ 1 } , ~ \pm \frac{ 1750 }{ 1 } , ~ \pm \frac{ 1875 }{ 1 } , ~ \pm \frac{ 2100 }{ 1 } , ~ \pm \frac{ 2500 }{ 1 } , ~ \pm \frac{ 2625 }{ 1 } , ~ \pm \frac{ 3000 }{ 1 } , ~ \pm \frac{ 3125 }{ 1 } , ~ \pm \frac{ 3500 }{ 1 } , ~ \pm \frac{ 3750 }{ 1 } , ~ \pm \frac{ 4200 }{ 1 } , ~ \pm \frac{ 4375 }{ 1 } , ~ \pm \frac{ 5000 }{ 1 } , ~ \pm \frac{ 5250 }{ 1 } , ~ \pm \frac{ 6250 }{ 1 } , ~ \pm \frac{ 7000 }{ 1 } , ~ \pm \frac{ 7500 }{ 1 } , ~ \pm \frac{ 8750 }{ 1 } , ~ \pm \frac{ 9375 }{ 1 } , ~ \pm \frac{ 10500 }{ 1 } , ~ \pm \frac{ 12500 }{ 1 } , ~ \pm \frac{ 13125 }{ 1 } , ~ \pm \frac{ 15000 }{ 1 } , ~ \pm \frac{ 15625 }{ 1 } , ~ \pm \frac{ 17500 }{ 1 } , ~ \pm \frac{ 18750 }{ 1 } , ~ \pm \frac{ 21000 }{ 1 } , ~ \pm \frac{ 21875 }{ 1 } , ~ \pm \frac{ 25000 }{ 1 } , ~ \pm \frac{ 26250 }{ 1 } , ~ \pm \frac{ 31250 }{ 1 } , ~ \pm \frac{ 35000 }{ 1 } , ~ \pm \frac{ 37500 }{ 1 } , ~ \pm \frac{ 43750 }{ 1 } , ~ \pm \frac{ 46875 }{ 1 } , ~ \pm \frac{ 52500 }{ 1 } , ~ \pm \frac{ 62500 }{ 1 } , ~ \pm \frac{ 65625 }{ 1 } , ~ \pm \frac{ 75000 }{ 1 } , ~ \pm \frac{ 87500 }{ 1 } , ~ \pm \frac{ 93750 }{ 1 } , ~ \pm \frac{ 105000 }{ 1 } , ~ \pm \frac{ 109375 }{ 1 } , ~ \pm \frac{ 125000 }{ 1 } , ~ \pm \frac{ 131250 }{ 1 } , ~ \pm \frac{ 175000 }{ 1 } , ~ \pm \frac{ 187500 }{ 1 } , ~ \pm \frac{ 218750 }{ 1 } , ~ \pm \frac{ 262500 }{ 1 } , ~ \pm \frac{ 328125 }{ 1 } , ~ \pm \frac{ 375000 }{ 1 } , ~ \pm \frac{ 437500 }{ 1 } , ~ \pm \frac{ 525000 }{ 1 } , ~ \pm \frac{ 656250 }{ 1 } , ~ \pm \frac{ 875000 }{ 1 } , ~ \pm \frac{ 1312500 }{ 1 } , ~ \pm \frac{ 2625000 }{ 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(75) = 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 - 75} $

$$ \frac{ -x^{3}+450x^{2}-63125x+2625000 }{ \color{blue}{ x - 75 } } = -x^{2}+375x-35000 $$

Polynomial $ -x^{2}+375x-35000 $ can be used to find the remaining roots.

$ \color{blue}{ -x^{2}+375x-35000 } $ 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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