Answer:
Therefore, using Descartes' Rule of Signs, we can conclude that the polynomial H(x) = 4x^4 - 5x^3 + 2x^2 - x + 5 has at most 2 positive roots and either 0 or 2 negative roots.
Explanation:
To use Descartes' Rule of Signs to determine the number of positive and negative roots of the polynomial H(x) = 4x^4 - 5x^3 + 2x^2 - x + 5, we need to follow these steps:
Step 1: Count the number of sign changes in the coefficients of H(x). A sign change occurs when the sign of a coefficient changes from positive to negative or from negative to positive as we go from one term to the next. In this case, we have:
4x^4 - 5x^3 + 2x^2 - x + 5
The coefficients are 4, -5, 2, -1, 5. There are two sign changes: from +4 to -5 and from -1 to +5.
Step 2: Determine the maximum possible number of positive roots of H(x) by counting the number of sign changes in the coefficients of H(x), or by subtracting an even integer from the number of terms with nonzero coefficients. In this case, there are two sign changes, so the maximum possible number of positive roots is 2.
Step 3: Determine the number of negative roots of H(x) by considering the polynomial H(-x) and following the same steps as in Steps 1 and 2. In this case, we have:
4(-x)^4 - 5(-x)^3 + 2(-x)^2 - (-x) + 5
= 4x^4 + 5x^3 + 2x^2 + x + 5
The coefficients are 4, 5, 2, 1, 5. There are no sign changes in this polynomial, so the number of negative roots is either 0 or an even number.
Therefore, using Descartes' Rule of Signs, we can conclude that the polynomial H(x) = 4x^4 - 5x^3 + 2x^2 - x + 5 has at most 2 positive roots and either 0 or 2 negative roots.