Without technology find all solutions to the following equation:

x⁷ sinx - 28x⁶ sinx - 5854x sinx + 4620sinx + 286x⁵ sinx - x⁷ + 28x⁶ -286x⁵ + 1302x⁴ -2351x³ - 406x² - 1302x⁴ sinx + 2351x³ sinx + 406x² sinx + 5854x - 4620 = 0

The only integer solutions are a set of 5 consecutive prime numbers.

Solution:

Let f(x) stand for the left side of the equation. Careful study of the function reveals that each power of x appears once with a coefficient and once with the negative of that coefficient and sinx. This means that f(x) = p(x) (sinx - 1) where
P(x) = x⁷ - 28x⁶ + 286x⁴ - 1302x³ + 406x² - 5854x + 4620.
The constant term 4620 = 2² · 3 · 5 ·7 ·11. Ignoring sign this is the product of all the roots of p(x) and so it seems 2 could be a root. Indeed, we can check that (x - 2) is a factor and therefore by the clue, p(x) = (x - 2)(x - 3)(x -5)(x - 7)(x - 11)q(x). Division yields q(x) = x² - 2. Therefore the solution set is {2, 3, 5, 7, 11, ¸, -Ã, ð/2 + 2nð}.

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