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Here is the problem: x squared plus 13x plus 40.

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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 40 equals 40. Find two numbers whose product is 40 and whose sum is b equals 13: 8 and 5.

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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 8 and negative 5.

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Write the complete factorization.

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So the answer is (x plus 5)(x plus 8).

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x squared plus 13x plus 40 factors completely over the rationals as (x plus 5)(x plus 8).

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Expanding (x plus 5)(x plus 8) gives back x squared plus 13x plus 40.