Let fx gx hx epsilon Fx such that fx does not equal 0 Prove

Let f(x), g(x), h(x), epsilon F[x] such that f(x) does not equal 0. Prove that: If f(x) | g(x) and f(x) | [g(x)^2 +h(x)], then f(x). Hint: g(x)^2 = g(x) * g(x)

Solution

given  f(x) | g(x)

=>g(x)=k.f(x)

=>g(x)²=k².f(x)²

=>g(x)²=l.f(x) eq1 where l=k².f(x)

and

f(x) | [g(x)^2 +h(x)]

=>[g(x)^2 +h(x)]=m.f(x) eq2

eq2-eq1

=>h(x)=(l-m)f(x)=p.f(x) where p=l-m

there fore f(x)|h(x)

then f(x). Hint: g(x)^2 = g(x) * g(x)