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)= -1x^4- 3x^3+8x^2+2x- 14
We can see above that there are two sign changes, ( -) to (+) and (+) to ( -). Therefore, there are either 2 or 0 positive real zeros.
Negative Real Zeros
Now consider P(- x).
P(- x)=-(- x)^4-3(- x)^3+8(- x)^2+2(- x)-14
⇕
P(- x)= - 1x^4+3x^3+8x^2- 2x- 14
We can see above that there are two sign changes, ( -) to (+) and (+) to ( -). Therefore, there are either 2 or 0 negative real zeros.
