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2.(a) How many distinct congruence classes are there modulo x^2+x+1 in Z5[x]? Can you find polynomials u(x) and v(x) in Z5[x] for which

2.(a) How many distinct congruence classes are there modulo x^2+x+1 in Z5[x]? Can you find polynomials u(x) and v(x) in Z5[x] for which 1=u(x)*(x+2)+v(x)*(x^2+x+1)?

(Hint: long division gives (x^2+x+1)=(x+2)*(x-1)+3, and 3 is invertible in Z5)

(b) Let d(x) be the greatest common divisor of x+1 and x^2+1 in the polynomial ring Z3[x]. Find polynomials u(x) and v(x) in Z3[x] for which d(x)=u(x)*(x+1)+v(x)*(x^2+1)

(Hint: you may find regarding the elements of Z3 as being -1,0, and +1 to be useful, instead of 0,1,2)

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