Respuesta :
Answer : C
[tex]f(x) = x^6 + x^5 + x^4 + 4x^3 - 12x^2 + 12[/tex]
For f(x), there will be 6 possible zeros
To get positive root we look at the sign changes in coefficients and count the sign changes in f(x)
We have two sign changes . Keep subtracting by 2 till we get 0
So positive roots are 2 or 0
Remaining 4 are complex. keep subtracting by 2 till we get 0
So complex roots are 4, 2 or 0
To get negative roots, we replace x with -x in f(x)
[tex]f(-x) = (-x)^6 + (-x)^5 + (-x)^4 + 4(-x)^3 - 12(-x)^2 + 12[/tex]
[tex]f-(x) = x^6 - x^5 + x^4 - 4x^3 - 12x^2 + 12[/tex]
To get negative root we look at the sign changes in coefficients and count the sign changes in f(-x)
We have two sign changes . Keep subtracting by 2 till we get 0
So negative roots are 2 or 0
Remaining 4 are complex. keep subtracting by 2 till we get 0
So complex roots are 4, 2 or 0
Answer is Positive: 4, 2, or 0; negative: 2 or 0; complex: 4, 2, or 0
