The coefficients of all terms are positive, so there are no sign changes between terms. According to Descartes’ Rule of Signs, this means there are no positive Real solutions.
We check for negative Real solutions by replacing x with –x and checking for sign changes. x^5 is changed to –x^5, but the remaining terms stay positive, so we have one sign change. We now have one sign change between terms, indicating there is at most one negative Real number solution.
Since the degree of the equation is 5, we expect 5 solutions. However, imaginary roots/solutions must come in pairs. (Otherwise you’d end up with “i” in the equation.) Therefore there cannot be 5 imaginary roots. There must be four imaginary roots and one negative Real root.
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