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Here is the problem: 3x squared minus 15x plus 12.

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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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Factor out the greatest common factor. Every term is divisible by 3.

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Use the AC method. Multiply a times c equals 1 times 4 equals 4. Find two numbers whose product is 4 and whose sum is b equals negative 5: negative 1 and negative 4.

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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 1 and 4.

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Write the complete factorization.

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So the answer is 3(x minus 4)(x minus 1).

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3x squared minus 15x plus 12 factors completely over the rationals as 3(x minus 4)(x minus 1).

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Expanding 3(x minus 4)(x minus 1) gives back 3x squared minus 15x plus 12.