polynomials depends on how much work you do. But I will assume that
you are expected to use Descartes rule of signs for these.
I will do the first one.
Look at the signs of the coefficients:
+ - - + - +
These change from + to - or - to + 4 times, so by DesCartes rule of
signs this can have either 4, 2 or 0 positive real roots.
Now replace x by -x in this to obtain:
-x^5 - 6x^4 + 3x^3 + 7x^2 + 8x + 1
so now we have:
- - + + + +
so there is 1 change from + to - or - to +, so in this case there must be
exactly 1 negative real root, (because if there are k changes in the signs
there can be k, k-2, ... , q, where q is zero if k is even and 1 if k is odd).
Now complex roots of a polynomial with real coefficients occur in conjugate
pairs, and the total number of roots is equal to the order of the polynomial
so here the order is 5 so there are a total of 5 roots altogether.
So here we have 1 negative root, 4, 2 or 0 positive real roots, with 0, 2 or 4
complex roots depending on the number positive roots. So in list form we
have the following possibilities:
1 negative, 4 positive 0 complex
1 negative, 2 positive 2 complex
1 negative, 0 positive 4 complex
The other problems are similar