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

Answer: 2

Let $2x^5+5x^4+10x^3+10x^2+10x+10=f(x)$
Now $f(-2)=-34$ and $f(-1)=3$
Hence $f(x)$ has a root in (-2,-1) Further
$f^{\prime }(x)=10x^4+20x^3+20x^2+20x+10$
$=10x^2\left[\left(x^2+\frac{1}{x^2}\right)+2\left(x+\frac{1}{x}\right)+20\right]$
$=10x^2\left[{\left(x+\frac{1}{x}+1\right)}^2+17\right]>0$
Hence $f(x)$ has only one real root, so $|a|=2$