$ \color{blue}{ 64x^{3}-48x^{2}+12x-344 } $ 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 ( 64 ) are 1 2 4 8 16 32 64 .The factors of the constant term (-344) are 1 2 4 8 43 86 172 344 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 8 } , ~ \pm \frac{ 1 }{ 16 } , ~ \pm \frac{ 1 }{ 32 } , ~ \pm \frac{ 1 }{ 64 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 4 } , ~ \pm \frac{ 2 }{ 8 } , ~ \pm \frac{ 2 }{ 16 } , ~ \pm \frac{ 2 }{ 32 } , ~ \pm \frac{ 2 }{ 64 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 2 } , ~ \pm \frac{ 4 }{ 4 } , ~ \pm \frac{ 4 }{ 8 } , ~ \pm \frac{ 4 }{ 16 } , ~ \pm \frac{ 4 }{ 32 } , ~ \pm \frac{ 4 }{ 64 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 8 }{ 2 } , ~ \pm \frac{ 8 }{ 4 } , ~ \pm \frac{ 8 }{ 8 } , ~ \pm \frac{ 8 }{ 16 } , ~ \pm \frac{ 8 }{ 32 } , ~ \pm \frac{ 8 }{ 64 } , ~ \pm \frac{ 43 }{ 1 } , ~ \pm \frac{ 43 }{ 2 } , ~ \pm \frac{ 43 }{ 4 } , ~ \pm \frac{ 43 }{ 8 } , ~ \pm \frac{ 43 }{ 16 } , ~ \pm \frac{ 43 }{ 32 } , ~ \pm \frac{ 43 }{ 64 } , ~ \pm \frac{ 86 }{ 1 } , ~ \pm \frac{ 86 }{ 2 } , ~ \pm \frac{ 86 }{ 4 } , ~ \pm \frac{ 86 }{ 8 } , ~ \pm \frac{ 86 }{ 16 } , ~ \pm \frac{ 86 }{ 32 } , ~ \pm \frac{ 86 }{ 64 } , ~ \pm \frac{ 172 }{ 1 } , ~ \pm \frac{ 172 }{ 2 } , ~ \pm \frac{ 172 }{ 4 } , ~ \pm \frac{ 172 }{ 8 } , ~ \pm \frac{ 172 }{ 16 } , ~ \pm \frac{ 172 }{ 32 } , ~ \pm \frac{ 172 }{ 64 } , ~ \pm \frac{ 344 }{ 1 } , ~ \pm \frac{ 344 }{ 2 } , ~ \pm \frac{ 344 }{ 4 } , ~ \pm \frac{ 344 }{ 8 } , ~ \pm \frac{ 344 }{ 16 } , ~ \pm \frac{ 344 }{ 32 } , ~ \pm \frac{ 344 }{ 64 } ~ $$

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(2) = 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 - 2} $

$$ \frac{ 64x^{3}-48x^{2}+12x-344 }{ \color{blue}{ x - 2 } } = 64x^{2}+80x+172 $$

Polynomial $ 64x^{2}+80x+172 $ can be used to find the remaining roots.

$ \color{blue}{ 64x^{2}+80x+172 } $ 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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