$$x^4-2x^3-33x^2+18x+216 = 0$$

$$ \begin{matrix}x_1 = 3 & x_2 = -3 & x_3 = -4 \\[1 em] x_4 = 6 & \\[1 em] \end{matrix} $$

$ \color{blue}{ x^{4}-2x^{3}-33x^{2}+18x+216 } $ is a polynomial of degree 4. 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 (216) are 1 2 3 4 6 8 9 12 18 24 27 36 54 72 108 216 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 54 }{ 1 } , ~ \pm \frac{ 72 }{ 1 } , ~ \pm \frac{ 108 }{ 1 } , ~ \pm \frac{ 216 }{ 1 } ~ $$

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} $

$$ \frac{ x^{4}-2x^{3}-33x^{2}+18x+216 }{ \color{blue}{ x - 3 } } = x^{3}+x^{2}-30x-72 $$

Polynomial $ x^{3}+x^{2}-30x-72 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{3}+x^{2}-30x-72 $

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

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