Question

For every $x\in R$, the polynomial $x^8-x^5+x^2-x+1$ is :

• A. positive
• B. never positive
• C. positive as well as negative
• D. negative

Answer 1

Answer: (A)

for $x\ge 1E=x^5\left(x^3-1\right)+(x-1)+1>0$
for $1<x<0,E=(1-x)+x^2\left(1-x^3\right)+x^8>0$
For $x<0$, all terms are positive $\Rightarrow >0$ Hence $A$