Question

Let $a$ be an integer such that all the real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $a,a+1$. Then, $|a|$ is equal to

Answer 1

Let, $f\left(x\right)=2x^5+5x^4+10x^3+10x^2+10x+10$
$\Rightarrow f^{\prime }\left(x\right)=10(x^4+2x^3+3x^2+2x+1)$
$=10x^2(x^2+\frac{1}{x^2}+2(x+\frac{1}{x})+3)$
$=10x^2({(x+\frac{1}{x})}^2+2(x+\frac{1}{x})+1)$
$=10x^2{((x+\frac{1}{x})+1)}^2>0;\forall x\in R$
$\therefore f\left(x\right)$ is strictly increasing function. Since it is an odd degree polynominal it will have exactly one real root.
Now
$f(-1)=3>0$
and $f(-2)=-64+80-80+40-20+10$
$=-34<0$
$\Rightarrow f\left(x\right)$ will have the root in the interval
$(-2,-1)\equiv (a,a+1)$
$\therefore |a|=2$