Let F be a field and let a ∈ F, with a ≠ 0 and f (X) = a_0 + a_1x + ... + a_nx^n ∈ F [X], with a_n ≠ 0.?
Show that if X - a divides f (X) , then X - (1 / a) divides a_n + a_(n-1)x + ... + a_0X^n
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Show that if X - a divides f (X) , then X - (1 / a) divides a_n + a_(n-1)x + ... + a_0X^n
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[Denote the latter polynomial by g(x)].
x - a | f(x)
⇒ f(a) = a(0) + ... + a(n).aⁿ = 0
⇒ g(1/a) = a(n) + ... + a(0).aⁿ = [a(0) + ... + a(n).aⁿ].(1/a)ⁿ = f(a).(1/a)ⁿ = 0
⇒ x - (1/a) | g(x).