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Here is the problem: x squared minus 3x minus 54.

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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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Use the AC method. Multiply a times c equals 1 times negative 54 equals negative 54. Find two numbers whose product is negative 54 and whose sum is b equals negative 3: 6 and negative 9.

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Factor by grouping. Group the first two and last two terms, factor each group, then factor out the common binomial. The roots are negative 6 and 9.

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Write the complete factorization.

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So the answer is (x minus 9)(x plus 6).

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x squared minus 3x minus 54 factors completely over the rationals as (x minus 9)(x plus 6).

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Expanding (x minus 9)(x plus 6) gives back x squared minus 3x minus 54.