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, | a | is equal to ______.

Step-by-step 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 ' x = 10 x 4 + 20 x 3 + 30 x 2 + 20 x + 10 = 10 x 2 x 2 + 2 x + 3 + 2 x + 1 x 2 = 10 x 2 x 2 + 1 x 2 + 2 x + 1 x + 3 = 10 x 2 x + 1 x 2 - 2 + 2 x + 1 x + 3 = 10 x 2 x + 1 x + 1 2 > 0 for all x belongs to R . f x is strictly increasing function. Since it is an odd degree polynomial it will have exactly one real root. Hence, f x has only one real root, so a = 2 .