The number of positive real roots of P(x)=0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less than that by an even number.
The number of negative real roots of P(x)=0 is either equal to the number of sign changes between consecutive coefficients of P(- x) or is less than that by an even number.
Positive Real Zeros
Consider the given polynomial P(x).
P(x)=5x^3+7x^2- 2x- 1
We can see above that there is only one sign change, (+) to ( -). Thus, there is only one positive real zero.
Negative Real Zeros
Now consider P(- x).
P(- x)=5(- x)^3+7(- x)^2-2(- x)-1
⇕
P(- x)= - 5x^3+7x^2+2x- 1
We can see that there are two sign changes, ( -) to (+) and (+) to ( -). Therefore, there are 2 or 0 negative real zeros.
