$$-432x^3+576x^2-213x+14 = 0$$

$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 12 } & x_2 = \dfrac{ 2 }{ 3 } & x_3 = \dfrac{ 7 }{ 12 } \end{matrix} $$

$ \color{blue}{ -432x^{3}+576x^{2}-213x+14 } $ 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 ( -432 ) are 1 2 3 4 6 8 9 12 16 18 24 27 36 48 54 72 108 144 216 432 .The factors of the constant term (14) are 1 2 7 14 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 3 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 6 } , ~ \pm \frac{ 1 }{ 8 } , ~ \pm \frac{ 1 }{ 9 } , ~ \pm \frac{ 1 }{ 12 } , ~ \pm \frac{ 1 }{ 16 } , ~ \pm \frac{ 1 }{ 18 } , ~ \pm \frac{ 1 }{ 24 } , ~ \pm \frac{ 1 }{ 27 } , ~ \pm \frac{ 1 }{ 36 } , ~ \pm \frac{ 1 }{ 48 } , ~ \pm \frac{ 1 }{ 54 } , ~ \pm \frac{ 1 }{ 72 } , ~ \pm \frac{ 1 }{ 108 } , ~ \pm \frac{ 1 }{ 144 } , ~ \pm \frac{ 1 }{ 216 } , ~ \pm \frac{ 1 }{ 432 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 3 } , ~ \pm \frac{ 2 }{ 4 } , ~ \pm \frac{ 2 }{ 6 } , ~ \pm \frac{ 2 }{ 8 } , ~ \pm \frac{ 2 }{ 9 } , ~ \pm \frac{ 2 }{ 12 } , ~ \pm \frac{ 2 }{ 16 } , ~ \pm \frac{ 2 }{ 18 } , ~ \pm \frac{ 2 }{ 24 } , ~ \pm \frac{ 2 }{ 27 } , ~ \pm \frac{ 2 }{ 36 } , ~ \pm \frac{ 2 }{ 48 } , ~ \pm \frac{ 2 }{ 54 } , ~ \pm \frac{ 2 }{ 72 } , ~ \pm \frac{ 2 }{ 108 } , ~ \pm \frac{ 2 }{ 144 } , ~ \pm \frac{ 2 }{ 216 } , ~ \pm \frac{ 2 }{ 432 } , ~ \pm \frac{ 7 }{ 1 } , ~ \pm \frac{ 7 }{ 2 } , ~ \pm \frac{ 7 }{ 3 } , ~ \pm \frac{ 7 }{ 4 } , ~ \pm \frac{ 7 }{ 6 } , ~ \pm \frac{ 7 }{ 8 } , ~ \pm \frac{ 7 }{ 9 } , ~ \pm \frac{ 7 }{ 12 } , ~ \pm \frac{ 7 }{ 16 } , ~ \pm \frac{ 7 }{ 18 } , ~ \pm \frac{ 7 }{ 24 } , ~ \pm \frac{ 7 }{ 27 } , ~ \pm \frac{ 7 }{ 36 } , ~ \pm \frac{ 7 }{ 48 } , ~ \pm \frac{ 7 }{ 54 } , ~ \pm \frac{ 7 }{ 72 } , ~ \pm \frac{ 7 }{ 108 } , ~ \pm \frac{ 7 }{ 144 } , ~ \pm \frac{ 7 }{ 216 } , ~ \pm \frac{ 7 }{ 432 } , ~ \pm \frac{ 14 }{ 1 } , ~ \pm \frac{ 14 }{ 2 } , ~ \pm \frac{ 14 }{ 3 } , ~ \pm \frac{ 14 }{ 4 } , ~ \pm \frac{ 14 }{ 6 } , ~ \pm \frac{ 14 }{ 8 } , ~ \pm \frac{ 14 }{ 9 } , ~ \pm \frac{ 14 }{ 12 } , ~ \pm \frac{ 14 }{ 16 } , ~ \pm \frac{ 14 }{ 18 } , ~ \pm \frac{ 14 }{ 24 } , ~ \pm \frac{ 14 }{ 27 } , ~ \pm \frac{ 14 }{ 36 } , ~ \pm \frac{ 14 }{ 48 } , ~ \pm \frac{ 14 }{ 54 } , ~ \pm \frac{ 14 }{ 72 } , ~ \pm \frac{ 14 }{ 108 } , ~ \pm \frac{ 14 }{ 144 } , ~ \pm \frac{ 14 }{ 216 } , ~ \pm \frac{ 14 }{ 432 } ~ $$

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 }{ 12 }) = 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}{ 12 x - 1 } $

$$ \frac{ -432x^{3}+576x^{2}-213x+14 }{ \color{blue}{ 12x - 1 } } = -36x^{2}+45x-14 $$

Polynomial $ -36x^{2}+45x-14 $ can be used to find the remaining roots.

$ \color{blue}{ -36x^{2}+45x-14 } $ 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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