Let a be an integer such that all the real roots of the polynomial
2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.

Let, $f(x) = 2{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 10x + 10$<br><br>$\Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)$<br><br>$$ = 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)$$<br><br>$$ = 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x + {1 \over x}} \right) + 1} \right)$$<br><br>$$ = 10{\left( {\left( {x + {1 \over x}} \right) + 1} \right)^2} &gt; 0;\forall x \in R$$<br><br>$\therefore$ f(x) is strictly increasing function. Since, it is an odd degree polynomial it will have exactly one real root.<br><br>Now, by observation.<br><br>$f( - 1) = 3 &gt; 0$<br><br>$f( - 2) = - 64 + 80 - 80 + 40 - 20 + 10$<br><br>$= - 34 &lt; 0$<br><br>$\Rightarrow f(x)$ has at least one root in $( - 2, - 1) \equiv (a,a + 1)$<br><br>$\Rightarrow a = - 2$ <br><br>$\Rightarrow$ |a| = - 2