$ \color{blue}{ -x^{3}+27x^{2}-240x+704 } $ 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 (704) are 1 2 4 8 11 16 22 32 44 64 88 176 352 704 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 11 }{ 1 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 22 }{ 1 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 44 }{ 1 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 88 }{ 1 } , ~ \pm \frac{ 176 }{ 1 } , ~ \pm \frac{ 352 }{ 1 } , ~ \pm \frac{ 704 }{ 1 } ~ $$

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(8) = 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 - 8} $

$$ \frac{ -x^{3}+27x^{2}-240x+704 }{ \color{blue}{ x - 8 } } = -x^{2}+19x-88 $$

Polynomial $ -x^{2}+19x-88 $ can be used to find the remaining roots.

$ \color{blue}{ -x^{2}+19x-88 } $ 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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