Use Descartes' rule of signs to describe the roots of the polynomial function.
m(x)=x^3+3x^2-18x-40
To use Descartes' rule of signs to describe the roots of the polynomial function, we need to examine the changes in sign of the coefficients.
1. First, list the coefficients of the polynomial function in descending order:
m(x) = x^3 + 3x^2 - 18x - 40
2. Count the number of sign changes in the coefficients. In this case, we have two sign changes.
3. The number of positive roots of the polynomial function is either equal to the number of sign changes or less than that by an even number. Since we have two sign changes, it means there are either two positive roots or zero positive roots.
4. Substitute -x for x in the polynomial function and repeat step 2. Now we have:
m(-x) = (-x)^3 + 3(-x)^2 - 18(-x) - 40
m(-x) = -x^3 + 3x^2 + 18x - 40
5. Count the number of sign changes in the coefficients of m(-x). In this case, we have zero sign changes.
6. The number of negative roots of the polynomial function is either equal to the number of sign changes or less than that by an even number. Since we have zero sign changes, it means there are either zero negative roots or two negative roots.
Therefore, according to Descartes' rule of signs, the roots of the polynomial function m(x) = x^3 + 3x^2 - 18x - 40 can be described as either two positive roots and zero negative roots or zero positive roots and two negative roots.