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Let a be an integer such that all the real roots of the... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

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 ___________.

Correct Answer:

2

Explanation

\text{ Let, } f(x) = 2{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 10x + 10

\Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)

= 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)

= 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x + {1 \over x}} \right) + 1} \right)

= 10{\left( {\left( {x + {1 \over x}} \right) + 1} \right)^2} > 0;\forall x \in R

\therefore f(x)

is strictly increasing function. Since, it is an odd degree polynomial it will have exactly one real root.

Now, by observation.

f( - 1) = 3 > 0

f( - 2) = - 64 + 80 - 80 + 40 - 20 + 10

= - 34 < 0

\Rightarrow f(x)

has at least one root in

( - 2, - 1) \equiv (a,a + 1)

\Rightarrow a = - 2

\Rightarrow |a| = - 2

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