$$(n^8-34n^7+478n^6-3580n^5+15289n^4-36706n^3+44712n^2-20160n)(n-9) = 0$$

$$ \begin{matrix}n_1 = 0 & n_2 = 1 & n_3 = 3 \\[1 em] n_4 = 4 & n_5 = 5 & n_6 = 6 \\[1 em] n_7 = 7 & n_8 = 8 & n_9 = 9 \end{matrix} $$

In order to solve $ \color{blue}{ x^{9}-43x^{8}+784x^{7}-7882x^{6}+47509x^{5}-174307x^{4}+375066x^{3}-422568x^{2}+181440x = 0 } $, first we need to factor our $ x $.

$$ x^{9}-43x^{8}+784x^{7}-7882x^{6}+47509x^{5}-174307x^{4}+375066x^{3}-422568x^{2}+181440x = x \left( x^{8}-43x^{7}+784x^{6}-7882x^{5}+47509x^{4}-174307x^{3}+375066x^{2}-422568x+181440 \right) $$

$ x = 0 $ is a root of multiplicity $ 1 $.

The remaining roots can be found by solving equation $ x^{8}-43x^{7}+784x^{6}-7882x^{5}+47509x^{4}-174307x^{3}+375066x^{2}-422568x+181440 = 0$.

$ \color{blue}{ x^{8}-43x^{7}+784x^{6}-7882x^{5}+47509x^{4}-174307x^{3}+375066x^{2}-422568x+181440 } $ is a polynomial of degree 8. 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 (181440) are 1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 27 28 30 32 35 36 40 42 45 48 54 56 60 63 64 70 72 80 81 84 90 96 105 108 112 120 126 135 140 144 160 162 168 180 189 192 210 216 224 240 252 270 280 288 315 320 324 336 360 378 405 420 432 448 480 504 540 560 567 576 630 648 672 720 756 810 840 864 945 960 1008 1080 1120 1134 1260 1296 1344 1440 1512 1620 1680 1728 1890 2016 2160 2240 2268 2520 2592 2835 2880 3024 3240 3360 3780 4032 4320 4536 5040 5184 5670 6048 6480 6720 7560 8640 9072 10080 11340 12096 12960 15120 18144 20160 22680 25920 30240 36288 45360 60480 90720 181440 . 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{ 9 }{ 1 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 14 }{ 1 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 21 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 28 }{ 1 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 35 }{ 1 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 42 }{ 1 } , ~ \pm \frac{ 45 }{ 1 } , ~ \pm \frac{ 48 }{ 1 } , ~ \pm \frac{ 54 }{ 1 } , ~ \pm \frac{ 56 }{ 1 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 63 }{ 1 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 70 }{ 1 } , ~ \pm \frac{ 72 }{ 1 } , ~ \pm \frac{ 80 }{ 1 } , ~ \pm \frac{ 81 }{ 1 } , ~ \pm \frac{ 84 }{ 1 } , ~ \pm \frac{ 90 }{ 1 } , ~ \pm \frac{ 96 }{ 1 } , ~ \pm \frac{ 105 }{ 1 } , ~ \pm \frac{ 108 }{ 1 } , ~ \pm \frac{ 112 }{ 1 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 126 }{ 1 } , ~ \pm \frac{ 135 }{ 1 } , ~ \pm \frac{ 140 }{ 1 } , ~ \pm \frac{ 144 }{ 1 } , ~ \pm \frac{ 160 }{ 1 } , ~ \pm \frac{ 162 }{ 1 } , ~ \pm \frac{ 168 }{ 1 } , ~ \pm \frac{ 180 }{ 1 } , ~ \pm \frac{ 189 }{ 1 } , ~ \pm \frac{ 192 }{ 1 } , ~ \pm \frac{ 210 }{ 1 } , ~ \pm \frac{ 216 }{ 1 } , ~ \pm \frac{ 224 }{ 1 } , ~ \pm \frac{ 240 }{ 1 } , ~ \pm \frac{ 252 }{ 1 } , ~ \pm \frac{ 270 }{ 1 } , ~ \pm \frac{ 280 }{ 1 } , ~ \pm \frac{ 288 }{ 1 } , ~ \pm \frac{ 315 }{ 1 } , ~ \pm \frac{ 320 }{ 1 } , ~ \pm \frac{ 324 }{ 1 } , ~ \pm \frac{ 336 }{ 1 } , ~ \pm \frac{ 360 }{ 1 } , ~ \pm \frac{ 378 }{ 1 } , ~ \pm \frac{ 405 }{ 1 } , ~ \pm \frac{ 420 }{ 1 } , ~ \pm \frac{ 432 }{ 1 } , ~ \pm \frac{ 448 }{ 1 } , ~ \pm \frac{ 480 }{ 1 } , ~ \pm \frac{ 504 }{ 1 } , ~ \pm \frac{ 540 }{ 1 } , ~ \pm \frac{ 560 }{ 1 } , ~ \pm \frac{ 567 }{ 1 } , ~ \pm \frac{ 576 }{ 1 } , ~ \pm \frac{ 630 }{ 1 } , ~ \pm \frac{ 648 }{ 1 } , ~ \pm \frac{ 672 }{ 1 } , ~ \pm \frac{ 720 }{ 1 } , ~ \pm \frac{ 756 }{ 1 } , ~ \pm \frac{ 810 }{ 1 } , ~ \pm \frac{ 840 }{ 1 } , ~ \pm \frac{ 864 }{ 1 } , ~ \pm \frac{ 945 }{ 1 } , ~ \pm \frac{ 960 }{ 1 } , ~ \pm \frac{ 1008 }{ 1 } , ~ \pm \frac{ 1080 }{ 1 } , ~ \pm \frac{ 1120 }{ 1 } , ~ \pm \frac{ 1134 }{ 1 } , ~ \pm \frac{ 1260 }{ 1 } , ~ \pm \frac{ 1296 }{ 1 } , ~ \pm \frac{ 1344 }{ 1 } , ~ \pm \frac{ 1440 }{ 1 } , ~ \pm \frac{ 1512 }{ 1 } , ~ \pm \frac{ 1620 }{ 1 } , ~ \pm \frac{ 1680 }{ 1 } , ~ \pm \frac{ 1728 }{ 1 } , ~ \pm \frac{ 1890 }{ 1 } , ~ \pm \frac{ 2016 }{ 1 } , ~ \pm \frac{ 2160 }{ 1 } , ~ \pm \frac{ 2240 }{ 1 } , ~ \pm \frac{ 2268 }{ 1 } , ~ \pm \frac{ 2520 }{ 1 } , ~ \pm \frac{ 2592 }{ 1 } , ~ \pm \frac{ 2835 }{ 1 } , ~ \pm \frac{ 2880 }{ 1 } , ~ \pm \frac{ 3024 }{ 1 } , ~ \pm \frac{ 3240 }{ 1 } , ~ \pm \frac{ 3360 }{ 1 } , ~ \pm \frac{ 3780 }{ 1 } , ~ \pm \frac{ 4032 }{ 1 } , ~ \pm \frac{ 4320 }{ 1 } , ~ \pm \frac{ 4536 }{ 1 } , ~ \pm \frac{ 5040 }{ 1 } , ~ \pm \frac{ 5184 }{ 1 } , ~ \pm \frac{ 5670 }{ 1 } , ~ \pm \frac{ 6048 }{ 1 } , ~ \pm \frac{ 6480 }{ 1 } , ~ \pm \frac{ 6720 }{ 1 } , ~ \pm \frac{ 7560 }{ 1 } , ~ \pm \frac{ 8640 }{ 1 } , ~ \pm \frac{ 9072 }{ 1 } , ~ \pm \frac{ 10080 }{ 1 } , ~ \pm \frac{ 11340 }{ 1 } , ~ \pm \frac{ 12096 }{ 1 } , ~ \pm \frac{ 12960 }{ 1 } , ~ \pm \frac{ 15120 }{ 1 } , ~ \pm \frac{ 18144 }{ 1 } , ~ \pm \frac{ 20160 }{ 1 } , ~ \pm \frac{ 22680 }{ 1 } , ~ \pm \frac{ 25920 }{ 1 } , ~ \pm \frac{ 30240 }{ 1 } , ~ \pm \frac{ 36288 }{ 1 } , ~ \pm \frac{ 45360 }{ 1 } , ~ \pm \frac{ 60480 }{ 1 } , ~ \pm \frac{ 90720 }{ 1 } , ~ \pm \frac{ 181440 }{ 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} $

$$ \frac{ x^{8}-43x^{7}+784x^{6}-7882x^{5}+47509x^{4}-174307x^{3}+375066x^{2}-422568x+181440 }{ \color{blue}{ x - 1 } } = x^{7}-42x^{6}+742x^{5}-7140x^{4}+40369x^{3}-133938x^{2}+241128x-181440 $$

Polynomial $ x^{7}-42x^{6}+742x^{5}-7140x^{4}+40369x^{3}-133938x^{2}+241128x-181440 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{7}-42x^{6}+742x^{5}-7140x^{4}+40369x^{3}-133938x^{2}+241128x-181440 $

When you get second degree polynomial use step-by-step quadratic equation solver to find two remaining roots.

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