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)=6x^4- 1x^3+5x^2- 1x+9
We can see above that there are four sign changes, (+) to ( -), ( -) to (+), (+) to ( -), and ( -) to (+). Therefore, there are either 4, 2, or 0 positive real zeros.
Negative Real Zeros
Now consider P(- x).
P(- x)=6(- x)^4-(- x)^3+5(- x)^2-(- x)+9
⇕
P(- x)=6x^4+1x^3+5x^2+1x+9
We can see that there are zero sign changes. Therefore, there are no negative real zeros.
