Question

Please help! 30 points!!

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

Side 1: 4y + 2y2 − 3

Side 2: −4 + 2y2 + 2y

Side 3: 4y2 − 3 + 2y

The perimeter of the quadrilateral is 22y3 + 10y2 + 10y − 17.

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

Part B: What is the length of the fourth side of the quadrilateral?

Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer.

Answer 1

Answer:

 (A) 8y^2 +8y -10

 (B) 22y^3 +2y^2 +2y -7

 (C) yes, for these polynomials. The result of adding or subtracting the polynomials in this problem is another polynomial, suggesting the set of polynomials is closed to addition and subtraction.

Step-by-step explanation:

A. (4y + 2y^2 − 3) + (−4 + 2y^2 + 2y) + (4y^2 − 3 + 2y)

 = y^2(2 +2 +4) +y(4 +2 +2) +(-3 -4 -3)

 = 8y^2 +8y -10 . . . . total length of sides 1, 2, 3

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B. The length of the 4th side is the difference between the perimeter and the sum of the other three sides:

 (22y^3 + 10y^2 + 10y − 17) -(8y^2 +8y -10)

 = 22y^3 +y^2(10 -8) +y(10 -8) +(-17 +10)

 = 22y^3 +2y^2 +2y -7 . . . . length of 4th side

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C. The result of adding or subtracting the polynomials in this problem is another polynomial, suggesting the set of polynomials is closed to addition and subtraction. An example is not a proof, but there is nothing about this example that would suggest a different conclusion.