State the number of positive real zeros, negative real zeros, and imaginary zeros for g(x)= 9x3 -7x2+10x-4.
Since there were 3 sign changes I have 3 positive real zeros.
For g(-x)= I have -9x3-7x2-10x-4. There were no sign changes so I have no negative real zeros.
The total number of positive real zeros therefore are 3 or 1, total number of negative real zeros is 0 and the total number of imaginary zeros is 0 or 2.
Is this correct?
Yes
Well-done!
Yes, your analysis is mostly correct. However, to determine the number of positive real zeros, you need to consider the number of sign changes in the coefficients of the function g(x). In this case, there were 2 sign changes (from 9 to -7 to 10), not 3. Therefore, the number of positive real zeros is 2 or 0.
For the number of negative real zeros, you correctly found that there were no sign changes when evaluating g(-x). Thus, the number of negative real zeros is 0.
Regarding the number of imaginary zeros, it is not possible to determine the exact number based on the given information. The number of imaginary zeros can only be determined if the degree of the polynomial equation is odd and the complex conjugate theorem is applicable. In this case, since the polynomial equation is of degree 3, it is possible to have 0 or 2 imaginary zeros.
So, to summarize:
- The number of positive real zeros is either 2 or 0.
- The number of negative real zeros is 0.
- The number of imaginary zeros is either 0 or 2.