Thread: complex polynom

hello, i have to determine the complex zero values of the function z^5+2z^3+z

ok here my try:

z (z^4+2z^2+1) = 0 => gives first zero pt. at 0
now substitution z^4 =u^2 and z^2 = u

now with pq formula i get this: -1+- 0

but what now ??
thanks in advance !

Let

By the remainder theorem you know that:

is a root of for some poloynomial

You did this correctly for to get

Then 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.