Question 1 (essay worth 10 points) (07.02 mc)
the lengths of three sides of a trapezoid are shown below: side 1: 11z2 + 4z − 6 side 2: 4z − 7 + 12z2 side 3: −8 + 13z2 + 4z the perimeter of the trapezoid is 7z3 + 42z2 − 15z + 1.

part a: what is the total length of sides 1, 2, and 3, of the trapezoid? (4 points)

part b: what is the length of the fourth side of the trapezoid? (3 points)

part c: do the answers for part a and part b show that the polynomials are closed under addition and subtraction? justify your answer. (3 points)

Part A:
Side 1 + side 2 + side 3 = 11z2 + 4z − 6 + 4z − 7 + 12z2  −8 + 13z2 + 4z
                                       = 11z2 + 12z2 + 13z2 + 4z + 4z + 4z - 6 - 7 - 8
                                       = 36z2 + 12z -21

Part B:
The parameter of the trapezoid is the summation of the lengths of its four sides, thus:
Side 4 = Parameter - (side 1 + side 2 + side 3)
           = Parameter - output of part A
           = 7z3 + 42z2 − 15z + 1 - 36z2 - 12z + 21
           = 7z3 + 6z2 - 27z +22

Part C:
For a polynomial to be closed under addition and subtraction operations, the performance of these operations on the polynomial should give a new polynomial.
Since the addition in Part A and the subtraction in Part B both resulted in new polynomials, thus, the answers of these two parts show that polynomials are closed under addition and subtraction.