In order to solve $ \color{blue}{ x^{9}-36x^{8}+546x^{7}-4536x^{6}+22449x^{5}-67284x^{4}+118124x^{3}-109584x^{2}+40320x = 0 } $, first we need to factor our $ x $.

$$ x^{9}-36x^{8}+546x^{7}-4536x^{6}+22449x^{5}-67284x^{4}+118124x^{3}-109584x^{2}+40320x = x \left( x^{8}-36x^{7}+546x^{6}-4536x^{5}+22449x^{4}-67284x^{3}+118124x^{2}-109584x+40320 \right) $$

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

The remaining roots can be found by solving equation $ x^{8}-36x^{7}+546x^{6}-4536x^{5}+22449x^{4}-67284x^{3}+118124x^{2}-109584x+40320 = 0$.

$ \color{blue}{ x^{8}-36x^{7}+546x^{6}-4536x^{5}+22449x^{4}-67284x^{3}+118124x^{2}-109584x+40320 } $ 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 (40320) are 1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 28 30 32 35 36 40 42 45 48 56 60 63 64 70 72 80 84 90 96 105 112 120 126 128 140 144 160 168 180 192 210 224 240 252 280 288 315 320 336 360 384 420 448 480 504 560 576 630 640 672 720 840 896 960 1008 1120 1152 1260 1344 1440 1680 1920 2016 2240 2520 2688 2880 3360 4032 4480 5040 5760 6720 8064 10080 13440 20160 40320 . 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{ 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{ 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{ 84 }{ 1 } , ~ \pm \frac{ 90 }{ 1 } , ~ \pm \frac{ 96 }{ 1 } , ~ \pm \frac{ 105 }{ 1 } , ~ \pm \frac{ 112 }{ 1 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 126 }{ 1 } , ~ \pm \frac{ 128 }{ 1 } , ~ \pm \frac{ 140 }{ 1 } , ~ \pm \frac{ 144 }{ 1 } , ~ \pm \frac{ 160 }{ 1 } , ~ \pm \frac{ 168 }{ 1 } , ~ \pm \frac{ 180 }{ 1 } , ~ \pm \frac{ 192 }{ 1 } , ~ \pm \frac{ 210 }{ 1 } , ~ \pm \frac{ 224 }{ 1 } , ~ \pm \frac{ 240 }{ 1 } , ~ \pm \frac{ 252 }{ 1 } , ~ \pm \frac{ 280 }{ 1 } , ~ \pm \frac{ 288 }{ 1 } , ~ \pm \frac{ 315 }{ 1 } , ~ \pm \frac{ 320 }{ 1 } , ~ \pm \frac{ 336 }{ 1 } , ~ \pm \frac{ 360 }{ 1 } , ~ \pm \frac{ 384 }{ 1 } , ~ \pm \frac{ 420 }{ 1 } , ~ \pm \frac{ 448 }{ 1 } , ~ \pm \frac{ 480 }{ 1 } , ~ \pm \frac{ 504 }{ 1 } , ~ \pm \frac{ 560 }{ 1 } , ~ \pm \frac{ 576 }{ 1 } , ~ \pm \frac{ 630 }{ 1 } , ~ \pm \frac{ 640 }{ 1 } , ~ \pm \frac{ 672 }{ 1 } , ~ \pm \frac{ 720 }{ 1 } , ~ \pm \frac{ 840 }{ 1 } , ~ \pm \frac{ 896 }{ 1 } , ~ \pm \frac{ 960 }{ 1 } , ~ \pm \frac{ 1008 }{ 1 } , ~ \pm \frac{ 1120 }{ 1 } , ~ \pm \frac{ 1152 }{ 1 } , ~ \pm \frac{ 1260 }{ 1 } , ~ \pm \frac{ 1344 }{ 1 } , ~ \pm \frac{ 1440 }{ 1 } , ~ \pm \frac{ 1680 }{ 1 } , ~ \pm \frac{ 1920 }{ 1 } , ~ \pm \frac{ 2016 }{ 1 } , ~ \pm \frac{ 2240 }{ 1 } , ~ \pm \frac{ 2520 }{ 1 } , ~ \pm \frac{ 2688 }{ 1 } , ~ \pm \frac{ 2880 }{ 1 } , ~ \pm \frac{ 3360 }{ 1 } , ~ \pm \frac{ 4032 }{ 1 } , ~ \pm \frac{ 4480 }{ 1 } , ~ \pm \frac{ 5040 }{ 1 } , ~ \pm \frac{ 5760 }{ 1 } , ~ \pm \frac{ 6720 }{ 1 } , ~ \pm \frac{ 8064 }{ 1 } , ~ \pm \frac{ 10080 }{ 1 } , ~ \pm \frac{ 13440 }{ 1 } , ~ \pm \frac{ 20160 }{ 1 } , ~ \pm \frac{ 40320 }{ 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} $

$$ \frac{ x^{8}-36x^{7}+546x^{6}-4536x^{5}+22449x^{4}-67284x^{3}+118124x^{2}-109584x+40320 }{ \color{blue}{ x - 1 } } = x^{7}-35x^{6}+511x^{5}-4025x^{4}+18424x^{3}-48860x^{2}+69264x-40320 $$

Polynomial $ x^{7}-35x^{6}+511x^{5}-4025x^{4}+18424x^{3}-48860x^{2}+69264x-40320 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{7}-35x^{6}+511x^{5}-4025x^{4}+18424x^{3}-48860x^{2}+69264x-40320 $

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

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