Question

What are the possible number of positive real, negative real, and complex zeros of
f(x) = -7x4 - 12x3 + 9x2 - 17x + 3?

Positive Real: 3 or 1
Negative Real: 1
Complex: 2 or 0

Positive Real: 3 or 1
Negative Real: 2 or 0
Complex: 1

Positive Real: 1
Negative Real: 3 or 1
Complex: 2 or 0

Positive Real: 4, 2 or 0
Negative Real: 1
Complex: 0 or 1 or 3

Answer 1

dang! ive been trying to do this but i cant. sorry

Answer 2

use descarets rule of sign ....

Answer 3

let x=1, and count the number of sign changes in the resulting summation, this is the possible number of positive roots

let x=-1, and count the number of sign changes in the resulting summation, this is the possible number of negative roots

the number of complex roots is a factor of how many pairs of complex roots can be removed from the real roots.

Answer 4

-7x4 - 12x3 + 9x2 - 17x + 3

let x be a positve number (i choose 1)

-7 - 12 + 9 - 17 + 3
| | | |
0 1 2 3

i count 3 sign changes, so there are at most 3 postive roots, but 2 of them might be complex so there is possibly only 1 postive root and 2 complex so far

try it when x=-1 and count the signs ...

Answer 5

from just working the positive roots, i can determine that only one option makes any sense.