Question

What are the possible number of positive real, negative real, and complex zeros of

f(x) = –7x4 – 12x3 + 9x2 – 17x + 3?
Answer:

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

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

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

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

Answer 1

The answer is C :)
It can be properly explained by others below or above me

Answer 2

f(x) = −7x^4 − 12x^3 + 9x^2 − 17x + 3 you will have to change the signs so,
f(−x) = −7x^4 + 12x^3 + 9x^2 + 17x + 3

For positive roots, count number of times coefficients of f(x) change sign:
− − + − +
3 times -----> 3 or 1 positive roots

For negative roots, count number of times coefficients of f(−x) change sign:
− + + + +
1 time -----> 1 negative root

Positive real: 3 or 1
Negative real: 1
Complex non-real: 0 or 2

The answer is C