$$(u^2-8u+12)(5u+35) = 0$$

$$ \begin{matrix}u_1 = 2 & u_2 = 6 & u_3 = -7 \end{matrix} $$

$ \color{blue}{ 5x^{3}-5x^{2}-220x+420 } $ 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 ( 5 ) are 1 5 .The factors of the constant term (420) are 1 2 3 4 5 6 7 10 12 14 15 20 21 28 30 35 42 60 70 84 105 140 210 420 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 5 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 5 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 3 }{ 5 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 4 }{ 5 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 5 }{ 5 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 6 }{ 5 } , ~ \pm \frac{ 7 }{ 1 } , ~ \pm \frac{ 7 }{ 5 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 10 }{ 5 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 12 }{ 5 } , ~ \pm \frac{ 14 }{ 1 } , ~ \pm \frac{ 14 }{ 5 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 15 }{ 5 } , ~ \pm \frac{ 20 }{ 1 } , ~ \pm \frac{ 20 }{ 5 } , ~ \pm \frac{ 21 }{ 1 } , ~ \pm \frac{ 21 }{ 5 } , ~ \pm \frac{ 28 }{ 1 } , ~ \pm \frac{ 28 }{ 5 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 30 }{ 5 } , ~ \pm \frac{ 35 }{ 1 } , ~ \pm \frac{ 35 }{ 5 } , ~ \pm \frac{ 42 }{ 1 } , ~ \pm \frac{ 42 }{ 5 } , ~ \pm \frac{ 60 }{ 1 } , ~ \pm \frac{ 60 }{ 5 } , ~ \pm \frac{ 70 }{ 1 } , ~ \pm \frac{ 70 }{ 5 } , ~ \pm \frac{ 84 }{ 1 } , ~ \pm \frac{ 84 }{ 5 } , ~ \pm \frac{ 105 }{ 1 } , ~ \pm \frac{ 105 }{ 5 } , ~ \pm \frac{ 140 }{ 1 } , ~ \pm \frac{ 140 }{ 5 } , ~ \pm \frac{ 210 }{ 1 } , ~ \pm \frac{ 210 }{ 5 } , ~ \pm \frac{ 420 }{ 1 } , ~ \pm \frac{ 420 }{ 5 } ~ $$

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} $

$$ \frac{ 5x^{3}-5x^{2}-220x+420 }{ \color{blue}{ x - 2 } } = 5x^{2}+5x-210 $$

Polynomial $ 5x^{2}+5x-210 $ can be used to find the remaining roots.

$ \color{blue}{ 5x^{2}+5x-210 } $ 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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