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)

We have the following sides of the trapezoids below:

Side 1: 11z² + 4z - 6
Side 2: 12z² + 4z - 7
Side 3: 13z² + 4z - 8

To find the total length of the three sides, we add the given expressions and simplify. Remember that when adding quadratic expressions, we can only add up like terms as shown below.

Side 1 + Side 2 + Side 3 = (11 + 12 + 13)z² + (4 + 4 + 4)z + (-6 + -7 + -8)
Side 1 + Side 2 + Side 3 = 36z² + 12z - 21 

Recall that the perimeter of a polygon is just the sum of its sides. Thus, we have

Perimeter = Side 1  + Side 2 + Side 3 + Side 4

So, if we have three sides and the perimeter, to find the fourth side, we just subtract the total of the three sides from the perimeter.

Side 4 = Perimeter - (Side 1 + Side 2 + Side 3)
Side 4 = (7z³ + 42z² - 15z + 1) - (36z² + 12z - 21)
Side 4 = 7z³ + 6z² - 27z + 22

From what could be seen, when adding and subtracting polynomials, the result will be a polynomial as well. thus, showing that polynomials are closed under addition and subtraction.

Part A: 36z² + 12z - 21 units
Part B: 7z³ + 6z² - 27z + 22 units
Part C: Yes

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