Question

Describe how you would analyze the zeros of the polynomial function f(x) = -3x5 - 8x4 +25x3 - 8x2 +x - 19 using Descartes' Rule of Signs and be sure to include the answer

Answer 1

-3x5 - 8x4 +25x3 - 8x2 +x - 19
Step 1:
List the signs of each term
--+-+-
Note 4 changes in sign (from - to + or from + to -)
So there is a maximum of 4 positive real roots (i.e. 0, 2 or 4)
Step 2:
Reverse the sign of odd-powers for the case of negative roots
+-----
So there is one change of sign, so there is a maximum of 1 real negative root.

For complex roots, the minimum number is 5-(4+1)=0.
Therefore, there can be 0, 2 or 4 complex roots, which is what will affect the number of positive or negative roots. In this case, it will only affect positive roots, since complex roots come in pairs.

[note: (not derived from Descartes' rule of signs)]
For this particular problem, there is only one real root, i.e. the negative root. The rest are all complex.

Answer 2

For more information, see
http://en.wikipedia.org/wiki/Descartes%27_rule_of_signs