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Here is the problem: 8x cubed minus 27.

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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 cubes: a cubed minus b cubed equals (a minus b)(a squared plus ab plus b squared).

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Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term negative 27 and q dividing the leading coefficient 8. Testing these candidates gives the root 3 over 2.

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Divide by (2x minus 3) using synthetic division at x equals 3 over 2.

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Check the remaining quadratic. Its discriminant is negative 108, negative, so it is irreducible over the rationals.

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Write the complete factorization.

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So the answer is (2x minus 3)(4x squared plus 6x plus 9).

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8x cubed minus 27 factors completely over the rationals as (2x minus 3)(4x squared plus 6x plus 9).

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Expanding (2x minus 3)(4x squared plus 6x plus 9) gives back 8x cubed minus 27.