Use Descarte's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation
-5r^(4)+6r^(3)+9r-15=0
To use Descartes' Rule of Signs, we need to examine the coefficients of the given equation.
The equation is: -5r^4 + 6r^3 + 9r - 15 = 0
1. Determine the number of positive roots:
Count the changes in sign of the coefficients as we move from left to right. For this equation, we have:
-5 (negative), 6 (positive), 9 (positive), -15 (negative)
There is only one sign change from negative to positive. Thus, there can be at most one positive root.
2. Determine the number of negative roots:
We apply the same process to the equation after multiplying each term by (-1) to eliminate any negative coefficients:
5r^4 - 6r^3 - 9r + 15 = 0
For this equation, we have:
5 (positive), -6 (negative), -9 (negative), 15 (positive)
Again, there is only one sign change from positive to negative. Therefore, there can be at most one negative root.
3. Determine the number of imaginary roots (non-real roots):
The total number of roots depends on the difference in the number of sign changes from one coefficient to another (0, 2, 4, etc.).
In the original equation, there are two sign changes (from -5 to 6, and from 9 to -15). Hence, there can be two or zero imaginary roots.
In summary, the possible number of roots for the equation -5r^4 + 6r^3 + 9r - 15 = 0 are as follows:
- At most one positive root.
- At most one negative root.
- Two or zero imaginary roots.