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 August 27th, 2013, 02:11 AM #1 Senior Member   Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2 Rewrite polynomial*2x as a sum of polynomial Hello there, I have a polynomial Pn(x) multiplied by 2x: $P_n(x)\, \cdot\, 2x$. I want to rewrite it as a sum like this: $\underbrace{P_n(x) + P_n(x) + ... + P_n(x)}_{\text{added 2\, \cdot\, x times}$ I realize I can do this at least one way: $\sum_{i= 1}^{2x}P_n(x)$. 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
 August 27th, 2013, 02:45 AM #2 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 879 Thanks: 60 Math Focus: सामान्य गणित Re: Rewrite polynomial*2x as a sum of polynomial Why do u want to write in more simpler form ?
 August 27th, 2013, 02:53 AM #3 Senior Member   Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2 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 $P_k$, In the case of $P_{k+1}$ I want to separate out the $P_k$ from $P_{k+1}$. The most difficult way I could achieve this was in trying rewrite my left-hand side as a sum of $P_n(x)$ 2x times, or some easily manipulable object. Thank you for your time. Kind regards, Marius
August 27th, 2013, 02:55 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: $P_n(x)\, \cdot\, 2x$. I want to rewrite it as a sum like this: $\underbrace{P_n(x) + P_n(x) + ... + P_n(x)}_{\text{added 2\, \cdot\, x times}$
There is something that bugs me about that but I don't really know...
lets say the grade of $P_n(x)$ is a. Then the grade of $P_n(x)*2x$ is equal to a+1.
But if you say the last is equal to the sum of $P_n(x)$ then the grade remains a. since a is not the same as a+1 That was wrong I think.

August 27th, 2013, 05:13 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: $P_n(x)\, \cdot\, 2x$. I want to rewrite it as a sum like this: $\underbrace{P_n(x) + P_n(x) + ... + P_n(x)}_{\text{added 2\, \cdot\, x times}$
x is the variable of the polynomial not a fixed whole number. So, what you're doing is not valid.

August 27th, 2013, 05:30 AM   #6
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Re: Rewrite polynomial*2x as a sum of polynomial

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 Because I want to prove an equation by induction.
May we see the complete problem?

August 27th, 2013, 07:11 AM   #7
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Quote:
 Originally Posted by king.oslo I want to separate out the $P_k$ from $P_{k+1}$.
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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