Let

By the remainder theorem you know that:

is a root of for some poloynomialYou did this correctly for to getThen by your substitution you found that , hence , so that we get.Therefore the roots are . The reason that you got 3 distinct roots instead of the 5 you might expect with an order 5 polynomial is that the factors and are both repeated in the (unique) factorisation of So really there are 5 roots, it's just that and are repeated.