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Here is the problem: x squared plus x minus 42.

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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 42 equals negative 42. Find two numbers whose product is negative 42 and whose sum is b equals 1: 7 and negative 6.

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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 7 and 6.

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Write the complete factorization.

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So the answer is (x minus 6)(x plus 7).

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x squared plus x minus 42 factors completely over the rationals as (x minus 6)(x plus 7).

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Expanding (x minus 6)(x plus 7) gives back x squared plus x minus 42.