# Condition for which the real roots of the polynomial is none

https://math.stackexchange.com/a/1302643/1021792 in this solution given it was being said as coefficients are all positive hence the equation $$x^6 + 4x^5 + 5x^4 + 4x^3 +2x +1 = 0$$ has no real number solution , but its in which cases always true that if all the coefficients are non negative then there will be no real roots at all ? As i think there are many counter examples like $$x^2 = 0$$ , $$x^3 +3x = 0$$ , i am guessing it might be true when the constant term is non zero , but was not able to show it .

• The answer you link refers to "solutions with $\color{red}{x \gt 0}$".
– dxiv
May 14 at 4:57
• If the coefficients are all positive and the constant term is positive then there is no way you can have a solution with $x>0$ since that would be a sum of positive numbers equal to zero... Solutions with $x<0$ would still be possible however i.e. $x+1=0$ May 14 at 4:59
• The most general way is Strum's theorem, which counts number of real zeros May 14 at 5:00
• Oh i see, now i understood , thanks @MichaelMorrow . Okay dxiv May 14 at 5:11
• Thats intersting @SeewooLee thanks May 14 at 5:12