- #1

DEMJR

- 14

- 0

I want to prove x^3 + a^2x + b^2 = 0 has one negative and two imaginary roots if b \not= 0.

I know that it cannot have any positive real roots because a > 0 and b > 0 will always be the case.

I believe I can prove this using Descarte's Rule of Signs (which is in the same chapter of this problem).

Using the rule of Signs I get f(-x) = -x^3 -a^2x + b^2 = 0 which implies one negative root. Thus we conclude that we have two imaginary roots since we have to have a total of 3 roots and none can be positive.

However, I do am not sure if this is exactly what is meant when it says to prove it. Does this suffice?

Also, how would I begin to prove this algebraically?

I know that it cannot have any positive real roots because a > 0 and b > 0 will always be the case.

I believe I can prove this using Descarte's Rule of Signs (which is in the same chapter of this problem).

Using the rule of Signs I get f(-x) = -x^3 -a^2x + b^2 = 0 which implies one negative root. Thus we conclude that we have two imaginary roots since we have to have a total of 3 roots and none can be positive.

However, I do am not sure if this is exactly what is meant when it says to prove it. Does this suffice?

Also, how would I begin to prove this algebraically?

Last edited: