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Irreducible-Polynomial-Examples3

Q1) Find the greatest common divisor (gcd)  d(x) , of the following polynomials over Q , the body of the rational numbers. Express d(x) as combination of polynomials i.e., d(x)=( f(x), g(x)) then d(x)=a(x)f(x)+b(x)g(x), where a(x), b(x) \(\epsilon\) Q[x] .

(i) \(x^{3}-6x+7\) and \(x+4\)

(ii) \(x^{2}-1\) and  \(2x^{7}-4x^{5}+2\)

(iii) \(3x^{2}+1\) and \(x^{6}+x^{4}+x+1\)

(iv) \(x^{3}-1\) and \(x^{7}-x^{4}+x^{3}-1\)


Solutions :

(i)  Let f(x)=\(x^{3}-6x+7\) and g(x)=x+4

      No common factors for f(x) and g(x).

     GCD of f(x) , g(x) is 1

   d(x)=1

   If d(x)= a(x)f(x) + b(x)g(x) then to find a(x), bx) such that  1=a(x)f(x) + b(x)g(x)

     \(x^{3}-6x+7\) =(x+4)(\(x^{2}-4x+10\))-33

      \(\frac{-1}{33}\)(\(x^{3}-6x+7\)) + \(\frac{1}{33}\)(x+4)(\(x^{2}-4x+10\)=1

      \(\frac{-1}{33}\)(\(x^{3}-6x+7\)) + (x+4)(\(\frac{x^{2}}{33}-\frac{4x}{33}+\frac{10}{33}\))=1

  Therefore     a(x)=\(\frac{-1}{33}\)

                        b(x)=\(\frac{x^{2}}{33}-\frac{4x}{33}+\frac{10}{33}\)



(ii) Let f(x)= \(x^{2}-1\) g(x)= \(2x^{7}-4x^{5}+2\)

    f(x) and g(x) has common factor x-1

   d(x)=x-1

   x-1 =a(x)f(x) + b(x)g(x) and to find a(x) and b(x)

  






      


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