Product of Roots of Sums of Products of Difference

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What is the product of all roots to the equation

++(x−1)(x−2)(x−3)+(x−2)(x−3)(x−4)(x−3)(x−4)(x−5)+(x−4)(x−5)(x−6)(x−5)(x−6)(x−7)+(x−6)(x−7)(x−8)=0?

  • Product of Roots of Sums of Products of Difference -

    If p(x) = an x^n + ...+a0,

    then,

    p(x) = an (x-r1)(x-r2)....(x-rn)

    where the rj are the roots of p(x). The product of all the roots is then related to p(0). Puting x = 0 in the above identity gives:

    p(0) = (-1)^n an Product from j = 1 to n of rj

    In your case, n = 6 and a6 = 2.

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