Descartes' rule

Question

Use Descartes' rule of signs to determine the possible number of positive real zeros and the negative real zeros for f(x)=x^4-9x^2-6x+4

Answer 1

Since f(x)=x^4-9x^2-6x+4 changes sign twice, by Descartes' rule of signs, there are maximum 2 positive real zeros .

 

 

 

Since f(-x)=x^4-9x^2+6x+4 changes sign twice, by Descartes' rule of signs, there are maximum 2 negative real zeros.

 

So, the combinations could be:

1) 2 positive real roots + 2 negative real roots

2) 2 positive real roots + 2 imaginary roots

3) 2 negative real roots + 2 imaginary roots

4) 4 imaginary roots