Problems see the attach file,

Thanks for the

1. Find polynomials ne), six} E Etc] with positive coefcients such that, if we 1write air} = Zea" all?) = Zvjl'j HIRE) = 21"ka then {pg}:=n and (lial are unin1odal sequences but {1"le is not unilnodal. In other words, nd two polynomials 1:Ivhose coeieients form unimodal sequences of positive integers, yet the sequence of coefcients for the product of the polynomials is not uni model. 2. Suppose all roots of HI] E ELI] are real numbers, and the nonzero ooefcients of Hz} are all positive. Prove that no root can be positive. 3. Consider the polynomial pk} = 1 +3I+512 +ra+5rd+335+er. (a) Is 11(3) logconcave? If so, verify all the necessary {in)equa.lities. If not, explain 1why not. [h] Determine the Fivector of 31(1)