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March 29, 2024, 01:36:32 am

Author Topic: Find the remainder  (Read 1925 times)  Share 

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Find the remainder
« on: January 19, 2008, 12:30:12 am »
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Find the remainder when is divided by . I'm a bit perplexed, thanks.

Ahmad

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Re: Find the remainder
« Reply #1 on: January 19, 2008, 01:07:08 am »
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Normally, when we divide integers, say 10 divided by 3, giving remainder of 1, we mean:



Where , is the quotient and is the remainder.





(This part is unnecessary, but you might find it interesting)
We extend this concept to the ring of polynomials with real coefficients, that is, . We require a concept of size to get a remainder with 'norm' smaller than the divisor, this is based on the degree of the polynomial. Anyway, to cut things short the ring of polynomials with real coefficients is a euclidean domain, so basically you can perform division.

So a polynomial can be divided by giving



In our case,

(Remainder is linear)

Subbing in yields







And our remainder is simply
« Last Edit: January 19, 2008, 08:19:43 am by Ahmad »
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Re: Find the remainder
« Reply #2 on: January 19, 2008, 01:33:29 am »
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Thanks a heap Ahmad, wow that's pretty ingenious!  :)

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Re: Find the remainder
« Reply #3 on: January 19, 2008, 03:19:15 pm »
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Oh hang on, this is an extension to the remainder theorem, I gotta pay more attention to how stuff is derived

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Re: Find the remainder
« Reply #4 on: January 19, 2008, 06:04:12 pm »
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Ahmad

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Re: Find the remainder
« Reply #5 on: January 19, 2008, 06:09:11 pm »
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You just sub in 1 and -1, you get etc
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midas_touch

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Re: Find the remainder
« Reply #6 on: January 20, 2008, 08:56:09 pm »
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Interesting question, are questions like these examinable on the current methods syllabus?
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Ahmad

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Re: Find the remainder
« Reply #7 on: January 20, 2008, 09:10:16 pm »
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I don't *think* so. :)
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Re: Find the remainder
« Reply #8 on: January 20, 2008, 10:35:00 pm »
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Normally, when we divide integers, say 10 divided by 3, giving remainder of 1, we mean:



Where , is the quotient and is the remainder.





(This part is unnecessary, but you might find it interesting)
We extend this concept to the ring of polynomials with real coefficients, that is, . We require a concept of size to get a remainder with 'norm' smaller than the divisor, this is based on the degree of the polynomial. Anyway, to cut things short the ring of polynomials with real coefficients is a euclidean domain, so basically you can perform division.

So a polynomial can be divided by giving



In our case,

(Remainder is linear)

Subbing in yields







And our remainder is simply

And this is why I look up to Ahmad, purely genius.
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