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    Irreducible Polynomials

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    Characters are:

    Q. Z(2)[x] (mod 2), Z[x], Q, Z(2), Z(3), Z(5), Z(9). all mods

    ? Decide, giving your reasons, which of the following polynomials is irreducible over . Factorize those polynomials that are reducible into a product of irreducible factors.

    a)
    b)
    c)

    ? Find, with justification, all monic irreducible cubics in .
    ? Find, with justification, a polynomial in that is irreducible over but not over .

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    https://brainmass.com/math/basic-calculus/irreducible-polynomials-and-factors-55750

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    1 Decide, giving your reasons, which of the following polynomials is irreducible over . Factorise those polynomials that are reducible into a product of irreducible factors.

    a)
    b)
    c)

    2 Find, with justification, all monic irreducible cubics in .
    3 Find, with justification, a polynomial in that is irreducible over but not over .

    Characters are: Q, Z(2)[x] (mod 2), Z[x], Q, Z(2), Z(3), Z(5), Z(9). all mods

    Solutions:

    1. a ) Consider the function , f (x) =

    Going by Descartes Rule of Sign Change , since there being no sign change in their co-efficients , we say that ... there is no positive zero's for this polynomial , f (x) .

    f (-x) = ... number of sign change = 3

    So , possible number of negative roots = 3 or 1

    We then use Rational Root Test .. to get the possible rational roots as .

    Since there is no positive root , only possible zero = - 1 . Root = x + 1

    Now from synthetic division ... we get the reduced equation as :
    f (x) =

    b ) Consider the function , ...

    Solution Summary

    Irreducible polynomials are investigated.

    $2.49

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