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Math Help - Binomial Theorem application?

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    Super Member fardeen_gen's Avatar
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    Binomial Theorem application?

    After several operations of differentiation and multiplying by (x + 1) performed in an arbitrary order the polynomial x^8 + x^7 is changed to ax + b. Prove that the difference between the integers a and b is always divisible by 49.
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    Quote Originally Posted by fardeen_gen View Post
    After several operations of differentiation and multiplying by (x + 1) performed in an arbitrary order the polynomial x^8 + x^7 is changed to ax + b. Prove that the difference between the integers a and b is always divisible by 49.
    Define D to be the differention operator, i.e.

    Df = \frac{df}{dx}.

    Let's say a polynomial p(x) of degree n is "acceptable" if
    D^{n-1} p = ax + b
    where a - b is divisible by 49. It is easy to
    verify that x^8 + x^7 is acceptable.

    It is an immediate consequence that if p is acceptable, then
    so is Dp. If we can show that (x+1) p
    is acceptable whenever p is acceptable, we will be done.

    So suppose p is an acceptable polynomial of degree n; i.e.
    D^{n-1}p = ax + b where a-b is divisible
    by 49. By the extended rule for differentiation of a product
    (which has a binomial-theorem-like look to it),

    D^n (x+1)p = (x+1)D^n p + \binom{n}{1} D (x+1) D^{n-1} p
    = (x+1) a + n (1) (ax + b)
    = (n+1)ax + a+nb

    and (n+1)a - (a+nb) = n(a - b), which is divisible
    by 49; so (x+1)p is acceptable. We're done.
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