descartes rule of signs to determine the possible number of positive and negative real zeros of: p(x)=x^5-x^4+x^3-x^2+x-5

Answer 1

To use Descartes rule check to see how many sign changes there are. For the given equation there are 5 sign changes which mean there are 5, 3 or 1 positive roots. (count down by 2). Now if you substitute a -x in for x check to see how many sign changes there are. x⁵ will become negative, x⁴ will stay negative, x³ will become negative, x² will stay negative, x will become negative and -5 will stay negative. Therefore there are no sign changes which means there are no negative roots. This means the possible roots are 5 positive and 0 negative or 3 positive, 0 negative and 2 imaginary or 1 positive, 0 negative and 4 imaginary roots.