Please help me.
Use Descartes' rule of signs to describe the roots for each polynomial function.
3. m(x)=x^3+3x^2-18x-40
1 positive, 2 or 0 negative
check:
http://www.wolframalpha.com/input/?i=x^3%2B3x^2-18x-40
To use Descartes' rule of signs to describe the roots of the polynomial function m(x) = x^3 + 3x^2 - 18x - 40, follow these steps:
Step 1: Count the number of sign changes in the coefficients.
- In this polynomial, we have the signs: +, +, -, -
- There are two sign changes from positive to negative.
Step 2: Find the number of positive roots.
- The number of positive roots can be equal to the number of sign changes or less by an even number.
- In this case, we have two sign changes, so the number of positive roots can be 2 or 0.
Step 3: Substitute (-x) for x and count the sign changes in the coefficients.
- m(-x) = (-x)^3 + 3(-x)^2 - 18(-x) - 40
= -x^3 + 3x^2 + 18x - 40
- In this polynomial, we have the signs: -, +, +, -
- There are two sign changes from negative to positive.
Step 4: Find the number of negative roots.
- The number of negative roots can be equal to the number of sign changes or less by an even number.
- In this case, we have two sign changes, so the number of negative roots can be 2 or 0.
Therefore, according to Descartes' rule of signs, the polynomial function m(x) = x^3 + 3x^2 - 18x - 40 can have either two positive roots or zero positive roots, and it can also have either two negative roots or zero negative roots.