Let f(x) belong to F[x], where F is a field. Let a be a zero of f(x) ofmultiplicity n, and write f(x) = (x -a)nq(x). If b ≠ a is a zero ofq(x), show that b has the same multiplicity as a zero of q(x) as itdoes for f(x).

Let f(x) belong to F[x], where F is a field. Let a be a zero of f(x) ofmultiplicity n, and write f(x) = (x -a)nq(x). If b ≠ a is a zero ofq(x), show that b has the same multiplicity as a zero of q(x) as itdoes for f(x).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 16E

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Question

Let f(x) belong to F[x], where F is a field. Let a be a zero of f(x) of
multiplicity n, and write f(x) = (x -a)ⁿq(x). If b ≠ a is a zero of
q(x), show that b has the same multiplicity as a zero of q(x) as it
does for f(x).

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