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Math Help - Divisibility

  1. #1
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    Divisibility

    Had the following question:

    If r is in Z and r is a non-zero solution of x^2+ax+b=0 (where a,b are in Z) prove that r divides b.

    I solved it the following way:
    Letting c and r be the roots of the polynomial.
    (x-c)(x-r)=0
    x^2+x(-r-c)+cr=0
    cr=b, which is the definition of r dividing b

    The question then goes on by saying:
    Determine for which natural numbers n the polynomial Pn(x)=(x^n)+(x^n-1)+....x+1 has integer roots and find them. (By an integer root we mean z in Z such that f(z)=0). Give two solutions for the problem
    1) By proving a suitable generalization of the already said theorem above.
    2) By using complex numbers and the polynomial x^(n+1)-1

    Yeah I have no idea what to do here. Any help appreciated.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by JaysFan31 View Post
    Had the following question:

    If r is in Z and r is a non-zero solution of x^2+ax+b=0 (where a,b are in Z) prove that r divides b.

    I solved it the following way:
    Letting c and r be the roots of the polynomial.
    (x-c)(x-r)=0
    x^2+x(-r-c)+cr=0
    cr=b, which is the definition of r dividing b

    The question then goes on by saying:
    Determine for which natural numbers n the polynomial Pn(x)=(x^n)+(x^n-1)+....x+1 has integer roots and find them. (By an integer root we mean z in Z such that f(z)=0). Give two solutions for the problem
    1) By proving a suitable generalization of the already said theorem above.
    2) By using complex numbers and the polynomial x^(n+1)-1

    Yeah I have no idea what to do here. Any help appreciated.
    1) Generalize your proof of the first theorem to show that if r is a non-zero solution of x^n + ax^{n-1} + ... + b = 0 that r divides b. (r,a,...,b in Z).

    Now, given the polynomial equation x^n + x^{n-1} + ... + 1 = 0 we know that for any non-zero r that solves the equation, r must divide 1. (And r is an integer.) Which integers do this? What then can you say about n?

    -Dan
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  3. #3
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    Quote Originally Posted by JaysFan31 View Post

    The question then goes on by saying:
    Determine for which natural numbers n the polynomial Pn(x)=(x^n)+(x^n-1)+....x+1 has integer roots and find them.
    You can factor,
    x^{n+1)-1=0 as,
    (x-1)(x^n+x^{n-1}+...+x+1)=0
    Now,
    The above equation can be solved by de Moiver's theorem.
    It has two intergral solutions x=1,x=-1 for n being even and 1, x=1 for n being odd. Thus, since x=1 is already a solution, thus the other is -1 if it exists and n+1 is even. Thus,
    x^n+x^{n-1}+...+x+1=0
    Has only other integral solution when n=>1 is odd and that solution is x=-1.
    Note, it cannot have any other solutions because it is impossible by de Moiver's theorem.
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  4. #4
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    Quote Originally Posted by JaysFan31 View Post
    Had the following question:

    If r is in Z and r is a non-zero solution of x^2+ax+b=0 (where a,b are in Z) prove that r divides b.

    I solved it the following way:
    Letting c and r be the roots of the polynomial.
    (x-c)(x-r)=0
    x^2+x(-r-c)+cr=0
    cr=b, which is the definition of r dividing b
    I do not favor such an approach because you never did show that it has roots in the integral domain Z.
    (Though you can work on the proof so that it can become acceptable).

    There is an eaiser way,
    by definition the evaluation homomorphism is zero thus,
    r^2+ar+b=0
    Thus,
    r(-r-a)=b
    We say that r divides b because there exists an integer,
    -r-b
    such that,
    r(-r-a)=b
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    I do not favor such an approach because you never did show that it has roots in the integral domain Z.
    Actually (x - c)(x - r) = x^2 + ax + b implies that c is an integer by the division algorithm. It boils down to the same condition that you gave: r(-r-a)=b. (Or were you objecting that this was the part of the argument that was missing?)

    -Dan
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  6. #6
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    Quote Originally Posted by topsquark View Post
    (Or were you objecting that this was the part of the argument that was missing?)
    There is nothing wrong with what he said it just does not look like the best way to approach the problem.
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