| Theorem 16.2 Division Algorithm for F[x] Let F be a field and let f(x), g(x) E F[x] with g(x) # 0. Then there exist unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x)q(x) + r(x) and either r(x) = 0 or deg r(x) < deg g(x). Corollary 2 Factor Theorem Let F be a field, a E F, and f(x) E F[x]. Then a is a zero of f(x) if and only if x – a is a factor of f(x).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Show that Corollary 2 of the Theorem is true over any commutative
ring with unity.

| Theorem 16.2 Division Algorithm for F[x]
Let F be a field and let f(x), g(x) E F[x] with g(x) # 0. Then
there exist unique polynomials q(x) and r(x) in F[x] such that f(x) =
g(x)q(x) + r(x) and either r(x) = 0 or deg r(x) < deg g(x).
Transcribed Image Text:| Theorem 16.2 Division Algorithm for F[x] Let F be a field and let f(x), g(x) E F[x] with g(x) # 0. Then there exist unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x)q(x) + r(x) and either r(x) = 0 or deg r(x) < deg g(x).
Corollary 2 Factor Theorem
Let F be a field, a E F, and f(x) E F[x]. Then a is a zero of f(x) if
and only if x – a is a factor of f(x).
Transcribed Image Text:Corollary 2 Factor Theorem Let F be a field, a E F, and f(x) E F[x]. Then a is a zero of f(x) if and only if x – a is a factor of f(x).
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