Answer:
Counting the number of sign changes in the function f(x)
The number of positive roots is equal to the number of sign changes in the function f(x) or less than that by an even integer.
Here, f(x)=3x^4-5x^3-x^2 has two sign changes (from + to -, and from - to +), so the number of positive roots is either 2 or 0.
Counting the number of sign changes in the function f(-x)
The number of negative roots is equal to the number of sign changes in the function f(-x) or less than that by an even integer.
Here, f(-x)=3x^4+5x^3-x^2 has one sign change (from - to +), so the number of negative roots is either 1 or 0.
Counting the number of non-real roots (complex roots)
The number of non-real roots (complex roots) is equal to the difference between the total number of roots (4 in this case) and the number of real roots (which we found above).
Therefore, the possible number of positive roots is 2 or 0, the possible number of negative roots is 1 or 0, and the possible number of complex roots is 2 or 4.
Explanation: