$$3x^4-8x^2+5 = 0$$

$$ \begin{matrix}x_1 = 1 & x_2 = -1 & x_3 = - \dfrac{\sqrt{ 15 }}{ 3 } \\[1 em] x_4 = \dfrac{\sqrt{ 15 }}{ 3 } & \\[1 em] \end{matrix} $$

$ \color{blue}{ 3x^{4}-8x^{2}+5 } $ 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 ( 3 ) are 1 3 .The factors of the constant term (5) are 1 5 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 3 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 5 }{ 3 } ~ $$

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(1) = 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 - 1} $

$$ \frac{ 3x^{4}-8x^{2}+5 }{ \color{blue}{ x - 1 } } = 3x^{3}+3x^{2}-5x-5 $$

Polynomial $ 3x^{3}+3x^{2}-5x-5 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ 3x^{3}+3x^{2}-5x-5 $

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

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