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Irreducible-Polynomial-Examples

Q1 ) Prove that P(x)= \(x^{4}-11x^{3}+48x^{2}-83x+58\) is irreducible in Q[x].

Solution:

    P(x)= \(x^{4}-11x^{3}+48x^{2}-83x+58\)

    To prove that P(x) has no Rational factors.

     Let us use Rational Theorem to find all possible rational zeros.

     factors of 58 (p)= {1,2,29, 58}

     factors of 1(q) = {1}

     All possible rational zeros are \(pm 1 , \pm2,\pm29,\pm58\)

      P(\(\pm1\))\(\neq 0\),   P(\(\pm2\))\(\neq 0\),   P(\(\pm29\))\(\neq 0\),   P(\(\pm58\))\(\neq 0\)

   Therefore no rational factors for P(x) .

   P(x)=1*(\(x^{4}-11x^{3}+48x^{2}-83x+58\))

   a(x)=1 which is constant polynomial, b(x)=P(x)

   Hence it is Irreducible over Q[x].


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