$$x^4+11x^3-57x^2-445x+1050 = 0$$

$$ \begin{matrix}x_1 = 6 & x_2 = -7 & x_3 = -5-5 \sqrt{ 2 } \\[1 em] x_4 = -5+5 \sqrt{ 2 } & \\[1 em] \end{matrix} $$

$ \color{blue}{ x^{4}+11x^{3}-57x^{2}-445x+1050 } $ 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 factor of the leading coefficient ( 1 ) is 1 .The factors of the constant term (1050) are 1 2 3 5 6 7 10 14 15 21 25 30 35 42 50 70 75 105 150 175 210 350 525 1050 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 7 }{ 1 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 14 }{ 1 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 21 }{ 1 } , ~ \pm \frac{ 25 }{ 1 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 35 }{ 1 } , ~ \pm \frac{ 42 }{ 1 } , ~ \pm \frac{ 50 }{ 1 } , ~ \pm \frac{ 70 }{ 1 } , ~ \pm \frac{ 75 }{ 1 } , ~ \pm \frac{ 105 }{ 1 } , ~ \pm \frac{ 150 }{ 1 } , ~ \pm \frac{ 175 }{ 1 } , ~ \pm \frac{ 210 }{ 1 } , ~ \pm \frac{ 350 }{ 1 } , ~ \pm \frac{ 525 }{ 1 } , ~ \pm \frac{ 1050 }{ 1 } ~ $$

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(6) = 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 - 6} $

$$ \frac{ x^{4}+11x^{3}-57x^{2}-445x+1050 }{ \color{blue}{ x - 6 } } = x^{3}+17x^{2}+45x-175 $$

Polynomial $ x^{3}+17x^{2}+45x-175 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{3}+17x^{2}+45x-175 $

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

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