$$0 = 8x^4-128$$

$$ \begin{matrix}x_1 = 2 & x_2 = -2 & x_3 = 2 i \\[1 em] x_4 = -2 i & \\[1 em] \end{matrix} $$

$ \color{blue}{ -8x^{4}+128 } $ is a polynomial of degree 4. 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 ( -8 ) are 1 2 4 8 .The factors of the constant term (128) are 1 2 4 8 16 32 64 128 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 8 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 4 } , ~ \pm \frac{ 2 }{ 8 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 2 } , ~ \pm \frac{ 4 }{ 4 } , ~ \pm \frac{ 4 }{ 8 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 8 }{ 2 } , ~ \pm \frac{ 8 }{ 4 } , ~ \pm \frac{ 8 }{ 8 } , ~ \pm \frac{ 16 }{ 1 } , ~ \pm \frac{ 16 }{ 2 } , ~ \pm \frac{ 16 }{ 4 } , ~ \pm \frac{ 16 }{ 8 } , ~ \pm \frac{ 32 }{ 1 } , ~ \pm \frac{ 32 }{ 2 } , ~ \pm \frac{ 32 }{ 4 } , ~ \pm \frac{ 32 }{ 8 } , ~ \pm \frac{ 64 }{ 1 } , ~ \pm \frac{ 64 }{ 2 } , ~ \pm \frac{ 64 }{ 4 } , ~ \pm \frac{ 64 }{ 8 } , ~ \pm \frac{ 128 }{ 1 } , ~ \pm \frac{ 128 }{ 2 } , ~ \pm \frac{ 128 }{ 4 } , ~ \pm \frac{ 128 }{ 8 } ~ $$

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} $

$$ \frac{ -8x^{4}+128 }{ \color{blue}{ x - 2 } } = -8x^{3}-16x^{2}-32x-64 $$

Polynomial $ -8x^{3}-16x^{2}-32x-64 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ -8x^{3}-16x^{2}-32x-64 $

When you get second degree polynomial use step-by-step quadratic equation solver to find two remaining roots.

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