12B Let F be a field and let ax bx cx dx and tx be polynomia

(12B) Let F be a field and let a(x), b(x), c(x), d(x) and t(x) be polynomials in F[x] with a(x) not the zero polynomial. Prove that if d(x) = gcd(b(x), c(x)) and s(x) = gcd(a(x)b(x), a(x)c(x)), then s(x) divides a(x)d(x).

Solution

a(x) is not zero polynomial

d(x) being gcd divides both b(x) and c(x).

Similarly gcd being s(x) divides ab and ac

This implies s(x) is a factor of either a(x) or gcd of c(x) and b(x)

If s(x) is a factor of a(x) then s(x) divides a(x) otherwise it divides d(x) gcd of b c

Hence it follows that s(x) divides a(x) d(x)