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Here is the problem: x cubed minus 8.

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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 8 and q dividing the leading coefficient 1. Testing these candidates gives the root 2.

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

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Check the remaining quadratic. Its discriminant is negative 12, 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 2)(x squared plus 2x plus 4).

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

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Expanding (x minus 2)(x squared plus 2x plus 4) gives back x cubed minus 8.