Let f(x) = axm + am-prm-1 + . . . + a, and g(x) = b,x"+ b,-px"-1 + ... + bo belong to Q[x] and suppose that f(x)g(x) belongs to Z[x]. Prove that a,b; is an integer for every i and j.
Let f(x) = axm + am-prm-1 + . . . + a, and g(x) = b,x"+ b,-px"-1 + ... + bo belong to Q[x] and suppose that f(x)g(x) belongs to Z[x]. Prove that a,b; is an integer for every i and j.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.2: Properties Of Division
Problem 52E
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Step 1
Given that,
This implies that .
Suppose, .
By Gauss' Lemma, since factors in , so there exists q,r such that:
and,
and,
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