For a polynomial g(x) with real coefficients, let m g denote the number of distinct real roots of g(x). Suppose S in the set of polynomials with real coefficients defined by S = {(x 2 - 1) 2 (a 0 + a 1 x + a 2 x 2 + a 3 x 3 ): a 0 , a 1 , a 2 , a 3 
R}.
For a polynomial f, let f’ and f” denote its first and second order derivatives, respectively. Then the minimum possible value of 
, where f 
S , is_____
Text Solution
Verified by Experts5
(5)
f = (x 2 - 1) 2 (a 0 + a 1 x + a 2 x 2 + a 3 x 3 )
= (x + l) 2 (x - 1) 2 (a 0 + a 1 x + a 2 x 2 + a 3 x 3 )
For min value of 
, a 0 + aix + a 2 x 2 + a 3 x 3 must have only 1 real root which is 1 or -1, then 
= 3 & 
= 2 
Answer = 5
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