Question

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)

Answer 1

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.