The Division Algorithm: need proof for "f(x) = q(x)g(x)+r(x) where r(x) is either..."
Is it possible to prove the Division Algorithm without using the Well-Ordering Principle? According to what I've read the Well-ordering principle is equivalent to the statement of the principle of mathematical induction. The well-ordered property says that any set of non-negative integers has a least element. And you can use this fact for an inductive proof of the division algorithm. But the thing is, to prove that this property is true, you need set theory; which I don't understand. Should I just accept this statement, namely that "any set of non-negative integers has a least element" as an axiom and simply move on? The reason I'm asking this is since I'm currently trying to understand a proof for the division algorithm, and I don't know any set theory. Or should I learn some basic set theory before I continue with polynomial long division?
I need a proof for:
\(\displaystyle f(x) = q(x)g(x) + r(x)\) where \(\displaystyle r(x)\) is either a zeropolynomial or deg \(\displaystyle r<\) deg \(\displaystyle g\)
Is it possible to prove the Division Algorithm without using the Well-Ordering Principle? According to what I've read the Well-ordering principle is equivalent to the statement of the principle of mathematical induction. The well-ordered property says that any set of non-negative integers has a least element. And you can use this fact for an inductive proof of the division algorithm. But the thing is, to prove that this property is true, you need set theory; which I don't understand. Should I just accept this statement, namely that "any set of non-negative integers has a least element" as an axiom and simply move on? The reason I'm asking this is since I'm currently trying to understand a proof for the division algorithm, and I don't know any set theory. Or should I learn some basic set theory before I continue with polynomial long division?
I need a proof for:
\(\displaystyle f(x) = q(x)g(x) + r(x)\) where \(\displaystyle r(x)\) is either a zeropolynomial or deg \(\displaystyle r<\) deg \(\displaystyle g\)
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