The lengths of three sides of a quadrilateral are shown below:

Side 1: 3y^2 + 2y − 6

Side 2: 3y − 7 + 4y^2

Side 3: −8 + 5y^2 + 4y

The perimeter of the quadrilateral is 4y3 + 18y2 + 16y − 26.

Part A: What is the total length of sides 1, 2, and 3 of the quadrilateral? (4 points)

Part B: What is the length of the fourth side of the quadrilateral? (4 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. (2 points)

Part A: To find the lengths of sides 1, 2, and 3, we need to add them together. We can do this by combining like terms (terms that have the same variables, or no variables).

(3y² + 2y − 6) + (3y − 7 + 4y²) + (−8 + 5y² + 4y)

We can now group them.

(3y² + 4y² + 5y²) + (2y + 3y + 4y) + (-6 - 7 - 8)

Now we simplify

12y² + 9y - 21

Part B: To find the length of the 4th side, we need to subtract the combined length of the 3 sides we know from the total length (perimeter).

(4y³ + 18y² + 16y − 26) - (12y² + 9y - 21)

Simplify, subtract like terms.

4y³ + (18y² - 12y²) + (16y - 9y) + (-26 + 21)

4y³ + 6y² + 7y - 5 is the length of the 4th side.

Part C (sorry for the bad explanation): A set of numbers is closed, or has closure, under a given operation if the result of the operation on any two numbers in the set is also in the set. For example, the set of real numbers is closed under addition, because adding any two real numbers results in another real number. Likewise, the real numbers are closed under subtraction, multiplication and division (by a nonzero real number), because performing these operations on two real numbers always yields another real number.

Polynomials are closed under the same operations as integers.