Let a be an integer such

Question:

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

Solution:

Let $2 x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+10 x+10=f(x)$

Now $f(-2)=-34$ and $f(-1)=3$

Hence $f(x)$ has a root in $(-2,-1)$

Further $f^{\prime}(x)=10 x^{4}+20 x^{3}+20 x^{2}+20 x+10$

$=10 x^{2}\left[\left(x^{2}+\frac{1}{x^{2}}\right)+2\left(x+\frac{1}{x}\right)+20\right]$

$=10 x^{2}\left[\left(x+\frac{1}{x}+1\right)^{2}+17\right]>0$

Hence $f(\mathrm{x})$ has only one real root, so $\mid \mathrm{al}=2$

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