Detailed solutions please Detailed solutions please 4 Let F

Detailed solutions please:

Detailed solutions please: 4. Let F be a field and let f(x), g(x), h(x), and d(x) be polynomials in F[x]. Prove that if d(x) = gcd(f(x), g(x)) and both f(x) and g(x) divide h(x), then prove that f(x)g(x) divides h(x)d(x).

Solution

if d(x) is the gcd of (f(x) , g(x))

=> that d(x) is a factor of both f(x) and g(x)

its also given that h(x)/f(x) and h(x)/g(x) both fractons given zero remainder

that is there is some common factor between h(x) and f(x) and h(x) and f(x) which cancles out from the both in both the two fractions.

so its true to say that

[h(x) * d(x)]/[f(x) * g(x)]

hence f(x)g(x) divides h(x)d(x)