$ \color{blue}{ 7290x^{4}-8100x^{3}+3240x^{2}-540x+30 } $ 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 ( 7290 ) are 1 2 3 5 6 9 10 15 18 27 30 45 54 81 90 135 162 243 270 405 486 729 810 1215 1458 2430 3645 7290 .The factors of the constant term (30) are 1 2 3 5 6 10 15 30 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 3 } , ~ \pm \frac{ 1 }{ 5 } , ~ \pm \frac{ 1 }{ 6 } , ~ \pm \frac{ 1 }{ 9 } , ~ \pm \frac{ 1 }{ 10 } , ~ \pm \frac{ 1 }{ 15 } , ~ \pm \frac{ 1 }{ 18 } , ~ \pm \frac{ 1 }{ 27 } , ~ \pm \frac{ 1 }{ 30 } , ~ \pm \frac{ 1 }{ 45 } , ~ \pm \frac{ 1 }{ 54 } , ~ \pm \frac{ 1 }{ 81 } , ~ \pm \frac{ 1 }{ 90 } , ~ \pm \frac{ 1 }{ 135 } , ~ \pm \frac{ 1 }{ 162 } , ~ \pm \frac{ 1 }{ 243 } , ~ \pm \frac{ 1 }{ 270 } , ~ \pm \frac{ 1 }{ 405 } , ~ \pm \frac{ 1 }{ 486 } , ~ \pm \frac{ 1 }{ 729 } , ~ \pm \frac{ 1 }{ 810 } , ~ \pm \frac{ 1 }{ 1215 } , ~ \pm \frac{ 1 }{ 1458 } , ~ \pm \frac{ 1 }{ 2430 } , ~ \pm \frac{ 1 }{ 3645 } , ~ \pm \frac{ 1 }{ 7290 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 2 }{ 2 } , ~ \pm \frac{ 2 }{ 3 } , ~ \pm \frac{ 2 }{ 5 } , ~ \pm \frac{ 2 }{ 6 } , ~ \pm \frac{ 2 }{ 9 } , ~ \pm \frac{ 2 }{ 10 } , ~ \pm \frac{ 2 }{ 15 } , ~ \pm \frac{ 2 }{ 18 } , ~ \pm \frac{ 2 }{ 27 } , ~ \pm \frac{ 2 }{ 30 } , ~ \pm \frac{ 2 }{ 45 } , ~ \pm \frac{ 2 }{ 54 } , ~ \pm \frac{ 2 }{ 81 } , ~ \pm \frac{ 2 }{ 90 } , ~ \pm \frac{ 2 }{ 135 } , ~ \pm \frac{ 2 }{ 162 } , ~ \pm \frac{ 2 }{ 243 } , ~ \pm \frac{ 2 }{ 270 } , ~ \pm \frac{ 2 }{ 405 } , ~ \pm \frac{ 2 }{ 486 } , ~ \pm \frac{ 2 }{ 729 } , ~ \pm \frac{ 2 }{ 810 } , ~ \pm \frac{ 2 }{ 1215 } , ~ \pm \frac{ 2 }{ 1458 } , ~ \pm \frac{ 2 }{ 2430 } , ~ \pm \frac{ 2 }{ 3645 } , ~ \pm \frac{ 2 }{ 7290 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 3 }{ 2 } , ~ \pm \frac{ 3 }{ 3 } , ~ \pm \frac{ 3 }{ 5 } , ~ \pm \frac{ 3 }{ 6 } , ~ \pm \frac{ 3 }{ 9 } , ~ \pm \frac{ 3 }{ 10 } , ~ \pm \frac{ 3 }{ 15 } , ~ \pm \frac{ 3 }{ 18 } , ~ \pm \frac{ 3 }{ 27 } , ~ \pm \frac{ 3 }{ 30 } , ~ \pm \frac{ 3 }{ 45 } , ~ \pm \frac{ 3 }{ 54 } , ~ \pm \frac{ 3 }{ 81 } , ~ \pm \frac{ 3 }{ 90 } , ~ \pm \frac{ 3 }{ 135 } , ~ \pm \frac{ 3 }{ 162 } , ~ \pm \frac{ 3 }{ 243 } , ~ \pm \frac{ 3 }{ 270 } , ~ \pm \frac{ 3 }{ 405 } , ~ \pm \frac{ 3 }{ 486 } , ~ \pm \frac{ 3 }{ 729 } , ~ \pm \frac{ 3 }{ 810 } , ~ \pm \frac{ 3 }{ 1215 } , ~ \pm \frac{ 3 }{ 1458 } , ~ \pm \frac{ 3 }{ 2430 } , ~ \pm \frac{ 3 }{ 3645 } , ~ \pm \frac{ 3 }{ 7290 } , ~ \pm \frac{ 5 }{ 1 } , ~ \pm \frac{ 5 }{ 2 } , ~ \pm \frac{ 5 }{ 3 } , ~ \pm \frac{ 5 }{ 5 } , ~ \pm \frac{ 5 }{ 6 } , ~ \pm \frac{ 5 }{ 9 } , ~ \pm \frac{ 5 }{ 10 } , ~ \pm \frac{ 5 }{ 15 } , ~ \pm \frac{ 5 }{ 18 } , ~ \pm \frac{ 5 }{ 27 } , ~ \pm \frac{ 5 }{ 30 } , ~ \pm \frac{ 5 }{ 45 } , ~ \pm \frac{ 5 }{ 54 } , ~ \pm \frac{ 5 }{ 81 } , ~ \pm \frac{ 5 }{ 90 } , ~ \pm \frac{ 5 }{ 135 } , ~ \pm \frac{ 5 }{ 162 } , ~ \pm \frac{ 5 }{ 243 } , ~ \pm \frac{ 5 }{ 270 } , ~ \pm \frac{ 5 }{ 405 } , ~ \pm \frac{ 5 }{ 486 } , ~ \pm \frac{ 5 }{ 729 } , ~ \pm \frac{ 5 }{ 810 } , ~ \pm \frac{ 5 }{ 1215 } , ~ \pm \frac{ 5 }{ 1458 } , ~ \pm \frac{ 5 }{ 2430 } , ~ \pm \frac{ 5 }{ 3645 } , ~ \pm \frac{ 5 }{ 7290 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 6 }{ 2 } , ~ \pm \frac{ 6 }{ 3 } , ~ \pm \frac{ 6 }{ 5 } , ~ \pm \frac{ 6 }{ 6 } , ~ \pm \frac{ 6 }{ 9 } , ~ \pm \frac{ 6 }{ 10 } , ~ \pm \frac{ 6 }{ 15 } , ~ \pm \frac{ 6 }{ 18 } , ~ \pm \frac{ 6 }{ 27 } , ~ \pm \frac{ 6 }{ 30 } , ~ \pm \frac{ 6 }{ 45 } , ~ \pm \frac{ 6 }{ 54 } , ~ \pm \frac{ 6 }{ 81 } , ~ \pm \frac{ 6 }{ 90 } , ~ \pm \frac{ 6 }{ 135 } , ~ \pm \frac{ 6 }{ 162 } , ~ \pm \frac{ 6 }{ 243 } , ~ \pm \frac{ 6 }{ 270 } , ~ \pm \frac{ 6 }{ 405 } , ~ \pm \frac{ 6 }{ 486 } , ~ \pm \frac{ 6 }{ 729 } , ~ \pm \frac{ 6 }{ 810 } , ~ \pm \frac{ 6 }{ 1215 } , ~ \pm \frac{ 6 }{ 1458 } , ~ \pm \frac{ 6 }{ 2430 } , ~ \pm \frac{ 6 }{ 3645 } , ~ \pm \frac{ 6 }{ 7290 } , ~ \pm \frac{ 10 }{ 1 } , ~ \pm \frac{ 10 }{ 2 } , ~ \pm \frac{ 10 }{ 3 } , ~ \pm \frac{ 10 }{ 5 } , ~ \pm \frac{ 10 }{ 6 } , ~ \pm \frac{ 10 }{ 9 } , ~ \pm \frac{ 10 }{ 10 } , ~ \pm \frac{ 10 }{ 15 } , ~ \pm \frac{ 10 }{ 18 } , ~ \pm \frac{ 10 }{ 27 } , ~ \pm \frac{ 10 }{ 30 } , ~ \pm \frac{ 10 }{ 45 } , ~ \pm \frac{ 10 }{ 54 } , ~ \pm \frac{ 10 }{ 81 } , ~ \pm \frac{ 10 }{ 90 } , ~ \pm \frac{ 10 }{ 135 } , ~ \pm \frac{ 10 }{ 162 } , ~ \pm \frac{ 10 }{ 243 } , ~ \pm \frac{ 10 }{ 270 } , ~ \pm \frac{ 10 }{ 405 } , ~ \pm \frac{ 10 }{ 486 } , ~ \pm \frac{ 10 }{ 729 } , ~ \pm \frac{ 10 }{ 810 } , ~ \pm \frac{ 10 }{ 1215 } , ~ \pm \frac{ 10 }{ 1458 } , ~ \pm \frac{ 10 }{ 2430 } , ~ \pm \frac{ 10 }{ 3645 } , ~ \pm \frac{ 10 }{ 7290 } , ~ \pm \frac{ 15 }{ 1 } , ~ \pm \frac{ 15 }{ 2 } , ~ \pm \frac{ 15 }{ 3 } , ~ \pm \frac{ 15 }{ 5 } , ~ \pm \frac{ 15 }{ 6 } , ~ \pm \frac{ 15 }{ 9 } , ~ \pm \frac{ 15 }{ 10 } , ~ \pm \frac{ 15 }{ 15 } , ~ \pm \frac{ 15 }{ 18 } , ~ \pm \frac{ 15 }{ 27 } , ~ \pm \frac{ 15 }{ 30 } , ~ \pm \frac{ 15 }{ 45 } , ~ \pm \frac{ 15 }{ 54 } , ~ \pm \frac{ 15 }{ 81 } , ~ \pm \frac{ 15 }{ 90 } , ~ \pm \frac{ 15 }{ 135 } , ~ \pm \frac{ 15 }{ 162 } , ~ \pm \frac{ 15 }{ 243 } , ~ \pm \frac{ 15 }{ 270 } , ~ \pm \frac{ 15 }{ 405 } , ~ \pm \frac{ 15 }{ 486 } , ~ \pm \frac{ 15 }{ 729 } , ~ \pm \frac{ 15 }{ 810 } , ~ \pm \frac{ 15 }{ 1215 } , ~ \pm \frac{ 15 }{ 1458 } , ~ \pm \frac{ 15 }{ 2430 } , ~ \pm \frac{ 15 }{ 3645 } , ~ \pm \frac{ 15 }{ 7290 } , ~ \pm \frac{ 30 }{ 1 } , ~ \pm \frac{ 30 }{ 2 } , ~ \pm \frac{ 30 }{ 3 } , ~ \pm \frac{ 30 }{ 5 } , ~ \pm \frac{ 30 }{ 6 } , ~ \pm \frac{ 30 }{ 9 } , ~ \pm \frac{ 30 }{ 10 } , ~ \pm \frac{ 30 }{ 15 } , ~ \pm \frac{ 30 }{ 18 } , ~ \pm \frac{ 30 }{ 27 } , ~ \pm \frac{ 30 }{ 30 } , ~ \pm \frac{ 30 }{ 45 } , ~ \pm \frac{ 30 }{ 54 } , ~ \pm \frac{ 30 }{ 81 } , ~ \pm \frac{ 30 }{ 90 } , ~ \pm \frac{ 30 }{ 135 } , ~ \pm \frac{ 30 }{ 162 } , ~ \pm \frac{ 30 }{ 243 } , ~ \pm \frac{ 30 }{ 270 } , ~ \pm \frac{ 30 }{ 405 } , ~ \pm \frac{ 30 }{ 486 } , ~ \pm \frac{ 30 }{ 729 } , ~ \pm \frac{ 30 }{ 810 } , ~ \pm \frac{ 30 }{ 1215 } , ~ \pm \frac{ 30 }{ 1458 } , ~ \pm \frac{ 30 }{ 2430 } , ~ \pm \frac{ 30 }{ 3645 } , ~ \pm \frac{ 30 }{ 7290 } ~ $$

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 }{ 3 }) = 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}{ 3 x - 1 } $

$$ \frac{ 7290x^{4}-8100x^{3}+3240x^{2}-540x+30 }{ \color{blue}{ 3x - 1 } } = 2430x^{3}-1890x^{2}+450x-30 $$

Polynomial $ 2430x^{3}-1890x^{2}+450x-30 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ 2430x^{3}-1890x^{2}+450x-30 $

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

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