MHB Is the Remainder of Polynomial $f(x)$ the Same for Two Different Divisors?

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The discussion centers on proving that the remainder of the polynomial f(x) = 2008 + 2007x + 2006x^2 + ... + 2x^2006 + x^2007 is identical when divided by the divisors x(x+1) and x(x+1)^2. Participants acknowledge a previous solution and express gratitude for contributions to the topic. The conversation highlights the mathematical approach to finding remainders in polynomial division. An additional solution is mentioned, indicating ongoing engagement with the problem. The focus remains on the equivalence of remainders for the specified divisors.
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Show that the remainder of the polynomial $f(x)=2008+2007x+2006x^2+\cdots+3x^{2005}+2x^{2006}+x^{2007}$ is the same upon division by $x(x+1)$ as upon division by $x(x+1)^2$.
 
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$f(x)=2008+2007x+2006x^2+\cdots+3x^{2005}+2x^{2006}+x^{2007}$1st we provide the premise

this shall have same remainder when divided by $x(x+1)$ and $x(x+1)^2$

provided this shall have same remainder when divided by $(x+1)$ and $(x+1)^2$

when we divide by (x+1)^2 we shall have a linear polynomial say m(x+1) + c and when we divide by (x+1) it shall be d and both are same if m= 0 and m+c = d

so if we convert
f(x) as a polynomial of (x+1) the coefficient of x should be zero

or f(x-1) should have coefficient of x to be zero

now we provide the solution based on premise

$f(x-1)=2008+2007(x-1)+2006(x-1)^2+\cdots+3(x-1)^{2005}+2(x-1)^{2006}+(x-1)^{2007}$

should have coefficient of x to be zero

the coefficient of x
$= 2007 + 2006 * (-2) + 2005 * 3 \cdots + 2 * 2006 - 2007 = 0$

hence proved
 
kaliprasad said:
$f(x)=2008+2007x+2006x^2+\cdots+3x^{2005}+2x^{2006}+x^{2007}$1st we provide the premise

this shall have same remainder when divided by $x(x+1)$ and $x(x+1)^2$

provided this shall have same remainder when divided by $(x+1)$ and $(x+1)^2$

when we divide by (x+1)^2 we shall have a linear polynomial say m(x+1) + c and when we divide by (x+1) it shall be d and both are same if m= 0 and m+c = d

so if we convert
f(x) as a polynomial of (x+1) the coefficient of x should be zero

or f(x-1) should have coefficient of x to be zero

now we provide the solution based on premise

$f(x-1)=2008+2007(x-1)+2006(x-1)^2+\cdots+3(x-1)^{2005}+2(x-1)^{2006}+(x-1)^{2007}$

should have coefficient of x to be zero

the coefficient of x
$= 2007 + 2006 * (-2) + 2005 * 3 \cdots + 2 * 2006 - 2007 = 0$

hence proved

Ops, I want to apologize to kaliprasad for not replying to your solution earlier...sorry, Kali!:(

Thanks for your participation and your great solution by the way. :cool:

I have another solution that I want to share here with you and MHB:

We have that

$\begin{align*}f(x)&=2008+1004x+1003(x^3+2x^2+x)+\cdots+3(x^{2003}+2x^{2002}+x^{2001})+2(x^{2005}+2x^{2004}+x^{2003})+(x^{2007}+2x^{2006}+x^{2005})\\&=x(x+1)^2(1003+\cdots+3x^{2000}+2x^{2002}+x^{2004})+(1004x+2008)\end{align*}$

from which the result follows with remainder $1004x+2008$
 
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