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Math Help - Divisible by 10^9 please help!

  1. #1
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    Divisible by 10^9 please help!

    Let be a polynomial with real coefficients. Show that there exists a nonzero polynomial with real coefficients such that has terms that are all of a degree divisible by .
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  2. #2
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    Quote Originally Posted by usagi_killer View Post
    Let be a polynomial with real coefficients. Show that there exists a nonzero polynomial with real coefficients such that has terms that are all of a degree divisible by .
    Here's one way to approach this problem. To get an idea of what is involved, start with a simpler problem.

    Let \color{blue}P(x) be a polynomial with real coefficients. Show that there exists a nonzero polynomial \color{blue}Q(x) with real coefficients such that \color{blue}P(x)Q(x) has terms that are all of a degree divisible by 2.

    A bit of thought shows that this is not too hard. Split P(x) into terms of even degree and terms of odd degree. Then we can write P(x) = A(x) + xB(x), where both A(x) and B(x) have all terms of even degree. If Q(x) = A(x)- xB(x) then P(x)Q(x) = (A(x) + xB(x))(A(x) - xB(x)) = (A(x))^2 - x^2(B(x))^2, which has all terms of even degree, as required. For convenience later on, define P_2(P(x)) = P(x)Q(x).

    Now try something a bit harder.

    Let \color{blue}P(x) be a polynomial with real coefficients. Show that there exists a nonzero polynomial \color{blue}Q(x) with real coefficients such that \color{blue}P(x)Q(x) has terms that are all of a degree divisible by 3.

    Let \omega = e^{2\pi i/3}, a complex cube root of unity. Check that if a+bx+cx^2 is a real quadratic polynomial then (a+bx+cx^2)(a+\omega bx+\omega^2 cx^2)(a+\omega^2 bx+\omega cx^2) has terms that are all of a degree divisible by 3. Also, the second and third of those three factors are complex conjugates of each other, so their product is a real polynomial.

    For a general polynomial P(x), we can write P(x) = A(x) + xB(x) + x^2C(x), where each of A(x), B(x) and C(x) has terms that are all of a degree divisible by 3. Let Q(x) = (A(x)+\omega xB(x)+\omega^2 x^2C(x))(A(x)+\omega^2 xB(x)+\omega x^2C(x)). Then P_3(P(x)) = P(x)Q(x) is a multiple of P(x) whose terms are all of a degree divisible by 3.

    The next part of the project is to do the same thing with 3 replaced by 5. You obviously have to start with a fifth complex root of unity, \omega=e^{2\pi i/5}, and the argument gets a bit tedious, so I'll skip it. The end result is a real polynomial Q(x) (a product of four complex factors) such that P_5(P(x)) = P(x)Q(x) is a multiple of P(x) whose terms are all of a degree divisible by 5.

    Now you can start to stitch things together. Notice that P_{10}(x) = P_5(P_2(P(x))) is a multiple of P(x) whose terms are all of a degree divisible by 10. Then inductively define P_{10^{k+1}}(P(x)) = P_{10}(P_{10^k}(P(x))). So finally, P_{10^9}(P(x)) is a multiple of P(x) whose terms are all of a degree divisible by 10^9.
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  3. #3
    Senior Member Shanks's Avatar
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    Wow! Amazing, A pertect application of the unit roots!
    How can you image this stuff?
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