$$(x+13)(13x^2+2x+1) = 0$$

$$ \begin{matrix}x_1 = -13 & x_2 = -\dfrac{ 1 }{ 13 }+\dfrac{ 2 \sqrt{ 3}}{ 13 }i & x_3 = -\dfrac{ 1 }{ 13 }-\dfrac{ 2 \sqrt{ 3}}{ 13 }i \end{matrix} $$

$ \color{blue}{ 13x^{3}+171x^{2}+27x+13 } $ 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 ( 13 ) are 1 13 .The factors of the constant term (13) are 1 13 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 13 } , ~ \pm \frac{ 13 }{ 1 } , ~ \pm \frac{ 13 }{ 13 } ~ $$

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(-13) = 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 + 13} $

$$ \frac{ 13x^{3}+171x^{2}+27x+13 }{ \color{blue}{ x + 13 } } = 13x^{2}+2x+1 $$

Polynomial $ 13x^{2}+2x+1 $ can be used to find the remaining roots.

$ \color{blue}{ 13x^{2}+2x+1 } $ 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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