I read in my math book today that you could prove the following theorem using long division.

For all polynomials p(x) and q(x) there exists polynomials k(x) and r(x) such that

p(x) = q(x)*k(x) + r(x)

where quotient k(x) and remainder r(x) is determined by p(x) and q(x) if the deg r(x) < deg of q(x).

My question is if anyone could show me the proof for the theorem above. Thanks.

Edit:

I've realised that my math book; which is an introductory course in mathematical analysis ( calculus 1) doesn't prove most of its theorems. Its suppose to be a book used for students in universities. But it seems to care more about maths applications than showing actual proofs. Now I don't care about its applications, I just wanna learn the underlying maths. And so I need to understand every single proof leading into calculus and beyond. Now does anyone have a recommendation for a book that proves everything that comes before and during calculus? I don't care if its hard, I just want to learn the proper way...

For all polynomials p(x) and q(x) there exists polynomials k(x) and r(x) such that

p(x) = q(x)*k(x) + r(x)

where quotient k(x) and remainder r(x) is determined by p(x) and q(x) if the deg r(x) < deg of q(x).

My question is if anyone could show me the proof for the theorem above. Thanks.

Edit:

I've realised that my math book; which is an introductory course in mathematical analysis ( calculus 1) doesn't prove most of its theorems. Its suppose to be a book used for students in universities. But it seems to care more about maths applications than showing actual proofs. Now I don't care about its applications, I just wanna learn the underlying maths. And so I need to understand every single proof leading into calculus and beyond. Now does anyone have a recommendation for a book that proves everything that comes before and during calculus? I don't care if its hard, I just want to learn the proper way...

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