$$3x^3-8x^2+12x+5 = 0$$

$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 3 } & x_2 = \dfrac{ 3 }{ 2 }+\dfrac{\sqrt{ 11 }}{ 2 }i & x_3 = \dfrac{ 3 }{ 2 }-\dfrac{\sqrt{ 11 }}{ 2 }i \end{matrix} $$

$ \color{blue}{ 3x^{3}-8x^{2}+12x+5 } $ 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 factors of the leading coefficient ( 3 ) are 1 3 .The factors of the constant term (5) are 1 5 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 3 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 5 }{ 3 } ~ $$

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(-\frac{ 1 }{ 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}{ 3 x + 1 } $

$$ \frac{ 3x^{3}-8x^{2}+12x+5 }{ \color{blue}{ 3x + 1 } } = x^{2}-3x+5 $$

Polynomial $ x^{2}-3x+5 $ can be used to find the remaining roots.

$ \color{blue}{ x^{2}-3x+5 } $ 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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