$$(t-3)^4 = 0$$

$$ t = 3 $$

$ \color{blue}{ x^{4}-12x^{3}+54x^{2}-108x+81 } $ 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 (81) are 1 3 9 27 81 . Then the Rational Roots Tests yields the following possible solutions:

$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 27 }{ 1 } , ~ \pm \frac{ 81 }{ 1 } ~ $$

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(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}{x - 3} $

$$ \frac{ x^{4}-12x^{3}+54x^{2}-108x+81 }{ \color{blue}{ x - 3 } } = x^{3}-9x^{2}+27x-27 $$

Polynomial $ x^{3}-9x^{2}+27x-27 $ can be used to find the remaining roots.

Use the same procedure to find roots of $ x^{3}-9x^{2}+27x-27 $

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

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