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Here is the problem: x to the power 4 minus 81.

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Factor a polynomial over the rational numbers.

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Write the polynomial in standard form.

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Recognise a special pattern. The remaining factor is a difference of squares: a squared minus b squared equals (a minus b)(a plus b).

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Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term negative 81 and q dividing the leading coefficient 1. Testing these candidates gives the roots negative 3, 3.

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Divide by (x plus 3) using synthetic division at x equals negative 3.

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Divide by (x minus 3) using synthetic division at x equals 3.

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Check the remaining quadratic. Its discriminant is negative 36, negative, so it is irreducible over the rationals.

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Write the complete factorization.

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So the answer is (x minus 3)(x plus 3)(x squared plus 9).

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x to the power 4 minus 81 factors completely over the rationals as (x minus 3)(x plus 3)(x squared plus 9).

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Expanding (x minus 3)(x plus 3)(x squared plus 9) gives back x to the power 4 minus 81.