Th Nov 5 13A Let F be a field and let axbxcx and px be polyn

Th, Nov 5: (13A) Let F be a field and let a(x),b(x),c(x) and p(x) be polynomials in F[x]. Prove that if gcd(a(x), c(x)) = IF and gcd(a(x), b(x)) = p(x), then gcd(a(x), b(x)c(x)) = p(x).

Solution

Given gcd( a(x), b(x) )= p(x)

means a(x)= p(x) *k(x)...(i)

and b(x)= p(x) *r(x)...(ii)

where k(x) and r(x) are in F[x] such that gcd( k(x), r(x))=1

multiply c(x) to equation (ii)

b(x)* c(x)= p(x) *r(x)* c(x)...(iii)

given gcd(a(x), c(x))=1...(iv)

means no factor is common in a(x) and c(x)

Using equation (i) and (iii) and (iv), we can clearly see that only p(x) is common in a(x) and b(x)*c(x)

Hence gcd( a(x), b(x)*c(x))=p(x)

proved:)