The lengths of three sides of a trapezoid are shown below:
side 1: 11z2 − 4z + 2

side 2: −2z + 3 + 12z2

side 3: 3 − 3z + 13z2

the perimeter of the trapezoid is 5z3 + 40z2 + 7z − 15.

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

part b: what is the length of the fourth side of the trapezoid?

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 +2 - 2z + 3 + 12z2 + 3 - 3z + 13z2
                                      = 11z2 + 12z2 + 13z2 - 4z - 2z - 3z + 2 + 3 + 3
                                      = 36z2 - 9z + 8
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
           = 5z3 + 40z2 + 7z - 15 - (36z2 - 9z +8)
           = 5z3 + 40z2 - 36z2 + 7z + 9z - 15 - 8
           = 5z3 + 4z2 + 16z - 23
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.

Yes, the correct answer is b.

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