I know a theorem which states: if $p(x)$ has integer coefficients and can be factored in $2$ polynomials with rational coefficients, than It can also be factored in $2$ polynomials with integer coefficients. But I can't answer this question.
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2The answer is yes because it’s a polynomial division with monic divisor. – lhf Nov 04 '24 at 09:45
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As mentioned above, with monic polynomial divisors, the usual polynomial long division algorithm works just fine, and needs only additions / subtractions and multiplications, hence never generates a non-integer coefficient. – Macavity Nov 04 '24 at 11:25
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1P.S.: You may also want to check https://math.stackexchange.com/questions/2140378/division-algorithm-for-polynomials-in-rx-where-r-is-a-commutative-ring-with-u?rq=1 and the like. – Macavity Nov 04 '24 at 11:26
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Thank you, you cleared all my doubts. – SimplyME10_ Nov 05 '24 at 05:50