Use descartes rule of sign to determine the possible number of positive real zeros and possible number of negative 6x^5-6x^4+7x^3-8

f(x) has 3 changes, so 3 or 1 + roots

f(-x) has no sign changes, so no - roots

check:

http://www.wolframalpha.com/input/?i=6x^5-6x^4%2B7x^3-8

To apply Descartes' Rule of Signs, we need to find the number of sign changes in the function, as well as the number of sign changes in its reverse order.

1. Start with the given polynomial: 6x^5 - 6x^4 + 7x^3 - 8.
2. Count the number of sign changes in the original polynomial by substituting positive values for x (excluding zero) and observing the sign. In this case, we have:
- For x = 1: (6)(1)^5 - (6)(1)^4 + (7)(1)^3 - 8 = 6 - 6 + 7 - 8 = -1 (negative value)
- For x = 2: (6)(2)^5 - (6)(2)^4 + (7)(2)^3 - 8 = 192 - 96 + 56 - 8 = 144 (positive value)
3. In this case, there is one sign change in the original polynomial.

4. Next, count the number of sign changes in the reverse order of the polynomial by substituting negative values for x (excluding zero) and observing the sign. In our case, we reverse the polynomial as: -8 + 7x^3 - 6x^4 + 6x^5.
- For x = -1: -8 + 7(-1)^3 - 6(-1)^4 + 6(-1)^5 = -8 - 7 - 6 - 6 = -27 (negative value)
- For x = -2: -8 + 7(-2)^3 - 6(-2)^4 + 6(-2)^5 = -8 + 56 + 96 - 192 = -48 (negative value)
5. In this case, there are two sign changes in the reversed polynomial.

6. According to Descartes' Rule of Signs, the possible number of positive real zeros is equal to the number of sign changes or less than that by an even positive integer. In this case, the possible number of positive real zeros is 1 or 3 (one sign change or less).
7. The possible number of negative real zeros is equal to the number of sign changes in the reversed polynomial or less than that by an even positive integer. In this case, the possible number of negative real zeros is 0 or 2 (two sign changes or less).

Therefore, based on Descartes' Rule of Signs, the given polynomial 6x^5 - 6x^4 + 7x^3 - 8 has either one or three positive real zeros and either zero or two negative real zeros.