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August 27th, 2013, 01:11 AM   #1
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Rewrite polynomial*2x as a sum of polynomial

Hello there,

I have a polynomial Pn(x) multiplied by 2x: . I want to rewrite it as a sum like this:

I realize I can do this at least one way:
. However, I want to know if there is a simpler way to write it, ideally an expression which is easy to manipulate algebraically.

Thank you for helping me learn mathematics.

Kind regards,
Marius
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August 27th, 2013, 01:45 AM   #2
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Re: Rewrite polynomial*2x as a sum of polynomial

Why do u want to write in more simpler form ?
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August 27th, 2013, 01:53 AM   #3
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Re: Rewrite polynomial*2x as a sum of polynomial

Because I want to prove an equation by induction.

I assume that the case is true for , In the case of I want to separate out the from . The most difficult way I could achieve this was in trying rewrite my left-hand side as a sum of 2x times, or some easily manipulable object.

Thank you for your time.

Kind regards,
Marius
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August 27th, 2013, 01:55 AM   #4
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Re: Rewrite polynomial*2x as a sum of polynomial

Quote:
Originally Posted by king.oslo
Hello there,

I have a polynomial Pn(x) multiplied by 2x: . I want to rewrite it as a sum like this:
There is something that bugs me about that but I don't really know...
lets say the grade of is a. Then the grade of is equal to a+1.
But if you say the last is equal to the sum of then the grade remains a. since a is not the same as a+1 That was wrong I think.
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August 27th, 2013, 04:13 AM   #5
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Re: Rewrite polynomial*2x as a sum of polynomial

Quote:
Originally Posted by king.oslo

I have a polynomial Pn(x) multiplied by 2x: . I want to rewrite it as a sum like this:
x is the variable of the polynomial not a fixed whole number. So, what you're doing is not valid.
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August 27th, 2013, 04:30 AM   #6
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Re: Rewrite polynomial*2x as a sum of polynomial

Quote:
Because I want to prove an equation by induction.
May we see the complete problem?
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August 27th, 2013, 06:11 AM   #7
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Quote:
Originally Posted by king.oslo
I want to separate out the from .
The original form allows you to do that. There's nothing wrong with writing it some other way, but it won't help you accomplish the proof by induction. As x is not the induction variable, it can be treated as a constant that need not be an integer (except, perhaps, if calculus is involved).
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