math

Show that a function f(n)= 2n^3+n^2+6n+3 always produces a number that is divisible by an odd number greater than 1, for any natural number, n.

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  1. let's factor it ...
    2n^3+n^2+6n+3
    = n^2(2n+1) + 3(2n+1)
    = (2n+1)(n^2 + 3)

    what kind of a number is 2n+1 ??

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  2. I do not know.

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  3. Well, if n is a whole number, then 2n will always be even.
    When you add 1 to any even, the result will be odd.
    So 2n+1 is always odd
    by writing it as
    2n^3+n^2+6n+3
    = (2n+1)(n^2 + 3)

    I have shown that one of the factors, namely (2n+1), will always be an odd number, thus
    2n^3+n^2+6n+3
    is always divisible by an odd number

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