1. Find polynomials 331:1], 91:} E ELI] with positive coefcients such that, if we write 331:3} = Elli-Ti 9111') || 'F'l 5 '71.. sirlq'lrl || :1 H II'...

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1. Find polynomials 331:1], 91:} E ELI] with positive coefficients such that, if we write 331:3} = Elli-Ti 9111') || 'F’l£5 '71.. sirlq‘lrl ||:1HII' then {pflLfl and (fl);=fl_ are uninroclsl sequences but {rflfla is not uninroclal. In otherwords? find two polynomials whose mefficients form uninrodal sequenoes of positive integers1 yet the sequence of coefficients for the product of the polynomials is not uni—modal. 2. Suppose all roots of f[1'] E 313:] are real numbers... and the nonzero coefficients of HI}are all positive. Prove that no root can be positive. 3. Consider the polynomialmix} = 1+3:s+5r“+t§ir3 +5m"+3::5+:::fi. [a] Is 111:3] log—concave? If so... 1verify all the necessary {in)equalities. If not? explainwhy not. [b] Determine the T—yector of 11(3).