Question

Determine how many zeros, how many real or complex, and find the roots for f(x) = x3 − 5x2 − 25x + 125.

Answer 1

We can write this as:-

P(x) = + x^3 - 5x^2 - 25x + 125
There are 2 changes of real sign so by Descartes Rule of  signs there are either 2  positive real roots or 0 positive roots.

P(-x)  = - x^3 - 5x^2 + 25x + 125 
There is just one change of sign  so  there is exactly 1 real negative root.

125 is a multiple of 5 so By rational root theorem  5 could be  a positive root.

P(5) =  125 - 125 - 125 + 125  = 0   so one zero is  5
if we divide   the polynomial by (x - 5) we get the quadratic
x^2 - 25 
(x + 5)(x - 5) = 0

x = 5,-5

so the roots are 5 (multiplicity 2)  and -5.
2 real positive zeroes  and one real negative zero