Use descartes rule of signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the following function: f

Question

Use descartes rule of signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the following function:

f(x)= 9x^7 + 3x^5 - 8x^3 + 7x -1

Answer 1

9, 3, -8, 7, -1

has 3 changes in sign, so there is a maximum 3 positive real zeros

take the original polynomial and change x to -x, 9x^7 -3x^5 -8x^3 +7x -1

becomes -9x^7 +3x^5 +8x^3 - 7x - 1 which has

coefficients -9, 3, 8, -7, -1 with 2 changes in sign, so there is a maximum 2 negative real zeros

Gauss would also note that the number of positive real zeros are 3 and if less than 3 then at most 1 positive real root, as the number decrease by 2