Determine the Possible Number of Real Roots f(x)=x^5+7x^4+2x^3+14x^2+x+7

To find the possible number of positive roots, look at the signs on the coefficients and count the number of times the signs on the coefficients change from positive to negative or negative to positive.

Since there are ${\displaystyle 0}$ sign changes from the highest order term to the lowest, there are at most ${\displaystyle 0}$ positive roots (Descartes\' Rule of Signs).

To find the possible number of negative roots, replace ${\displaystyle x}$ with ${\displaystyle -x}$ and repeat the sign comparison.

Since there are ${\displaystyle 5}$ sign changes from the highest order term to the lowest, there are at most ${\displaystyle 5}$ negative roots (Descartes\' Rule of Signs). The other possible numbers of negative roots are found by subtracting off pairs of roots (e.g. ${\displaystyle 5-2}$).