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Math Help - polynomials

  1. #1
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    polynomials

    1. a polynomial f(x) leaves remainder 15 and (2x+1) when divided by (x-3) and (x-1)^2 respectively.then,what is the remainder when it is divided by (x-3){(x-1)^2}

    2. find all polynomials P satisfying p(x+1)=p(x)+2x+1

    3. if all the roots of the polynomial equation
    x^4-4x^3+ax^2+bx+1=0 are positive real numbers,show that all the roots are equal.

    please help!

    question 2:
    i have observed that p(x)=x^2 satisfies the given condition.do we now have to substitute p(x) in the condition ?
    how should we substitute?
    Last edited by mr fantastic; September 4th 2010 at 02:40 PM. Reason: Merged.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by earthboy View Post

    2. find all polynomials P satisfying p(x+1)=p(x)+2x+1
    Suppose:

    p(x)=a_nx^n+a_{n-1}x^{n-1}+..._a_1x+a_0

    with a_n \ne 0, is such a polynomial. Then:

    p(x+1)=a_n(x+1)^n+a_{n-1}(x+1)^{n-1}+..._a_1(x+1)+a_0

    ...,,.... =a_nx^n + (na_n+a_{n-1})x^{n-1}+ ...

    which implies either that  (na_n+a_{n-1})=a_{n-1} (which is not possible as a_n \ne 0 ) or that n=2 and a_2=1

    etc

    CB
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