$$x^3+3x^2+2x = 50616$$

$$ \begin{matrix}x_1 = 36 & x_2 = -\dfrac{ 39 }{ 2 }+\dfrac{\sqrt{ 4103 }}{ 2 }i & x_3 = -\dfrac{ 39 }{ 2 }-\dfrac{\sqrt{ 4103 }}{ 2 }i \end{matrix} $$

$ \color{blue}{ x^{3}+3x^{2}+2x-50616 } $ 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 (-50616) are 1 2 3 4 6 8 9 12 18 19 24 36 37 38 57 72 74 76 111 114 148 152 171 222 228 296 333 342 444 456 666 684 703 888 1332 1368 1406 2109 2664 2812 4218 5624 6327 8436 12654 16872 25308 50616 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 19 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 37 }{ 1 } , ~ \pm \frac{ 38 }{ 1 } , ~ \pm \frac{ 57 }{ 1 } , ~ \pm \frac{ 72 }{ 1 } , ~ \pm \frac{ 74 }{ 1 } , ~ \pm \frac{ 76 }{ 1 } , ~ \pm \frac{ 111 }{ 1 } , ~ \pm \frac{ 114 }{ 1 } , ~ \pm \frac{ 148 }{ 1 } , ~ \pm \frac{ 152 }{ 1 } , ~ \pm \frac{ 171 }{ 1 } , ~ \pm \frac{ 222 }{ 1 } , ~ \pm \frac{ 228 }{ 1 } , ~ \pm \frac{ 296 }{ 1 } , ~ \pm \frac{ 333 }{ 1 } , ~ \pm \frac{ 342 }{ 1 } , ~ \pm \frac{ 444 }{ 1 } , ~ \pm \frac{ 456 }{ 1 } , ~ \pm \frac{ 666 }{ 1 } , ~ \pm \frac{ 684 }{ 1 } , ~ \pm \frac{ 703 }{ 1 } , ~ \pm \frac{ 888 }{ 1 } , ~ \pm \frac{ 1332 }{ 1 } , ~ \pm \frac{ 1368 }{ 1 } , ~ \pm \frac{ 1406 }{ 1 } , ~ \pm \frac{ 2109 }{ 1 } , ~ \pm \frac{ 2664 }{ 1 } , ~ \pm \frac{ 2812 }{ 1 } , ~ \pm \frac{ 4218 }{ 1 } , ~ \pm \frac{ 5624 }{ 1 } , ~ \pm \frac{ 6327 }{ 1 } , ~ \pm \frac{ 8436 }{ 1 } , ~ \pm \frac{ 12654 }{ 1 } , ~ \pm \frac{ 16872 }{ 1 } , ~ \pm \frac{ 25308 }{ 1 } , ~ \pm \frac{ 50616 }{ 1 } ~ $$

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(36) = 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 - 36} $

$$ \frac{ x^{3}+3x^{2}+2x-50616 }{ \color{blue}{ x - 36 } } = x^{2}+39x+1406 $$

Polynomial $ x^{2}+39x+1406 $ can be used to find the remaining roots.

$ \color{blue}{ x^{2}+39x+1406 } $ 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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