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MATHMAA

# 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)