$$-x^3+27x^2-180x+324 = 0$$
Answer
Explanation
$ \color{blue}{ -x^{3}+27x^{2}-180x+324 } $ 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 (324) are 1 2 3 4 6 9 12 18 27 36 54 81 108 162 324 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 54 }{ 1 } , ~ \pm \frac{ 81 }{ 1 } , ~ \pm \frac{ 108 }{ 1 } , ~ \pm \frac{ 162 }{ 1 } , ~ \pm \frac{ 324 }{ 1 } ~ $$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(3) = 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 - 3} $
$$ \frac{ -x^{3}+27x^{2}-180x+324 }{ \color{blue}{ x - 3 } } = -x^{2}+24x-108 $$Polynomial $ -x^{2}+24x-108 $ can be used to find the remaining roots.
$ \color{blue}{ -x^{2}+24x-108 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.