$$0 = -8x^3+1$$

Answer

$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 2 } & x_2 = -\dfrac{ 1 }{ 4 }+\dfrac{\sqrt{ 3 }}{ 4 }i & x_3 = -\dfrac{ 1 }{ 4 }-\dfrac{\sqrt{ 3 }}{ 4 }i \end{matrix} $$

Explanation

$$ \begin{aligned} 0 &= -8x^3+1&& \text{move all terms to the left hand side } \\[1 em]0+8x^3-1 &= 0&& \text{simplify left side} \\[1 em]8x^3-1 &= 0&& \\[1 em] \end{aligned} $$

$ \color{blue}{ 8x^{3}-1 } $ 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 ( 8 ) are 1 2 4 8 .The factor of the constant term ( -1 ) is 1 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 8 } ~ $$

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 }{ 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}{ 2 x - 1 } $

$$ \frac{ 8x^{3}-1 }{ \color{blue}{ 2x - 1 } } = 4x^{2}+2x+1 $$

Polynomial $ 4x^{2}+2x+1 $ can be used to find the remaining roots.

$ \color{blue}{ 4x^{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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