Let R[r] be the set of all expressions 00 a = do+ajx+azx += i=0 where a¡ E R for all nonnegative integers i. Informally, an element of R[x] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[r], analogous to the definitions for R[x] in the notes. Given a,b e R[x], your defi- nitions should indicate what each coefficient of the sum a+b and product ab is. "(b) Let f = ao+ajx+…+a„x" be a polynomial. I can treat f as an element of R*] by defining an+1,&n+2;+… all to equal 0. This shows that R[x] CR[x]. If you had already proved that R[x] was a ring, how could you use this fact to help you prove R] is a ring? [

A 3rd year student has 75 credits and a cumulative GPA of 2.88. This semester she is registered for 15 credits and hopes to raise his cumulative GPA to 3.00 and make the Dean’s List.   Is it possible? What semester GPA will she need to accomplish that? Show your calculations or post a screenshot from your online GPA grade calculator.

2. Your prank war has continued to accelerate. You decide to cover your friend's car with post- it notes. How many post-it notes would you have to buy if each post-it note package contains 500 sticky notes at 3 inches x 3 inches in size. Your friend drives a 2021 Jeep Wrangler. (Hint: start by finding the dimensions of the jeep).

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