$$2x^4-17x^3+38x^2-68x+120 = 0$$

$$ \begin{matrix}x_1 = \dfrac{ 5 }{ 2 } & x_2 = 6 & x_3 = 2 i \\[1 em] x_4 = -2 i & \\[1 em] \end{matrix} $$

$ \color{blue}{ 2x^{4}-17x^{3}+38x^{2}-68x+120 } $ 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 ( 2 ) are 1 2 .The factors of the constant term (120) are 1 2 3 4 5 6 8 10 12 15 20 24 30 40 60 120 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 3 }{ 2 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 2 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 5 }{ 2 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 6 }{ 2 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 8 }{ 2 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 10 }{ 2 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 12 }{ 2 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 15 }{ 2 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 20 }{ 2 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 24 }{ 2 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 30 }{ 2 } , ~ \pm \frac{ 40 }{ 1 } , ~ \pm \frac{ 40 }{ 2 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 60 }{ 2 } , ~ \pm \frac{ 120 }{ 1 } , ~ \pm \frac{ 120 }{ 2 } ~ $$

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{ 5 }{ 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 - 5 } $

$$ \frac{ 2x^{4}-17x^{3}+38x^{2}-68x+120 }{ \color{blue}{ 2x - 5 } } = x^{3}-6x^{2}+4x-24 $$

Polynomial $ x^{3}-6x^{2}+4x-24 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{3}-6x^{2}+4x-24 $

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

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