Results 1 to 2 of 2

Math Help - f(x)

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    152

    f(x)

    Determine all polynomials f(x)

    satisfying


    (x-243) f(3x)  = 243 (x-1)f(x)


    for all x.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member TheAbstractionist's Avatar
    Joined
    Apr 2009
    Posts
    328
    Thanks
    1
    When x=1 the RHS = 0 so the LHS must be 0 as well; thus f(3)=0.

    Substituting x=3 then gives RHS = 0 again, so LHS must be 0 \implies\,f(9)=0.

    Continuing this way, we get f(3)=f(9)=f(27)=f(81)=f(243)=0. Hence f(x) must have these five roots and so we have

    f(x)\ =\ (x-3)(x-9)(x-27)(x-81)(x-243)g(x)

    where g(x) is some polynomial.

    \therefore\ f(3x)\,=\,(3x-3)(3x-9)(3x-27)(3x-81)(3x-243)g(3x)

    =\,243(x-1)(x-3)(x-9)(x-27)(x-81)g(3x)

    \implies\ (x-243)f(3x)\,=\,243(x-1)(x-3)(x-9)(x-27)(x-81)(x-243)g(3x) = 243(x-1)f(x)

    Thus the polynomial g(x) must satisfy g(x)=g(3x) for all x and so must be a constant function. Hence

    \fbox{$f(x)~=~a(x-3)(x-9)(x-27)(x-81)(x-243)$}

    where a is a real constant.
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum