Only do this direction 1Proove if fafa0 then fxa2q for some

Only do this direction!!!

1.Proove if f(a)=f\'(a)=0 then f=(x-a)^2q for some q in F[x].

Let F be a field, a F, and f F[x]. Show that f(a) = f\'(a) = 0 if and only if f = (x - a)^2q for some q F[x], Here, f\' refers to the derivative of f. The product rule from calculus says (p middot q)\' = p\' middot q + p middot q\'-

Solution

If f(a) is zero it means (x-a) is factor of f.

If f\'(a) is zero then (x-a) is also factor of f\'

This means (x-a)^2 is factor of f.

Taking q as a constant that exist in field f

f(a) =f\'(a) =0 if and only if

(x-a)^2q

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