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Math Help - need help solving this function....

  1. #1
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    need help solving this function....

    A polynomial f(x) satisfies the equation f(x)+(x+1)3=2f(x+1). Find f(10).
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  2. #2
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    Re: need help solving this function....

    Write f(x) as a_3x^3+a_2x^2+a_1x+a_0 (you may also add a_5x^5+a_4x^4 or higher degrees), write the equation f(x)+(x+1)^3=2f(x+1) in terms of a_0, ..., a_3 and equate the coefficients of the corresponding monomials. You'll get a system of four linear equations in a_0, ..., a_3.
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  3. #3
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    Re: need help solving this function....

    ok, emakarov has the idea...
    try, but if you want to learn the full technique here it is

    Spoiler:
    your question: find f(10) if
    2f(x+1)-f(x)=3(x+1)
    now, you have stated that f(x) is a polynomial, I am assuming it is a quadratic polynomial (this dosen't always work though...but school problems are generally that way )
    so let f(x)=ax^2+bx+c
    so f(x+1)=a(x+1)^2+b(x+1)+c
    now according to your equation,
    2(a(x+1)^2+b(x+1)+c)-ax^2-bx-c=3x+3
    \implies ax^2+4ax+2a+bx+2b+c=3x+3 (you will get this by expanding and simplifying the above equation)
    now you can see that the right hand side has no x^2 in it, so a must be equal to 0 in order to equate it.
    if a=0, the equation shortens to bx+2b+c=3x+3
    we know the equation to be true for all values of x
    so put x=0, it becomes 2b+c=3
    put x=-1, it becomes b+c=0 or b=-c and so 2b+c=-2c+c=-c=3 or c=-3
    so we get c=-3,b=3,a=0 and our function f(x) becomes 3x-3.if f(x)=3x-3, \boxed{f(10)=3*10-3=27}
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