The lengths of two sides of a triangle are shown. Side 1: 3x2 − 2x − 1 Side 2: 9x + 2x2 − 3 The perimeter of the triangle is 5x3 + 4x2 − x − 3. Part A: What is the total length of the two sides, 1 and 2, of the triangle? Show your work.(4 points) Part B: What is the length of the third side of the triangle? Show your work. (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: 5·x²+7·x-4

Part B: 5·x³-x²-8·x+1

Part C: The polynomials for Part A and Part B show that the polynomials are closed under addition and subtraction

Step-by-step explanation:

We are given the length of side 1 as 3·x² - 2·x - 1

Side 2 = 9·x + 2·x² - 3

The perimeter of the triangle = 5·x³ + 4·x² - x - 3

Part A: The total length of the two sides is;

3·x² - 2·x - 1 + 9·x + 2·x² - 3 = 5·x²+7·x-4

Part B: Given that the perimeter = 5·x³ + 4·x² - x - 3

Therefore, the length of side 3 = Perimeter -length of (side 1 + side 2)

= 5·x³ + 4·x² - x - 3 - (5·x²+7·x-4) = 5·x³-x²-8·x+1

Part C: Whereby a polynomial is closed in an operation that produces another polynomial and the answers to both Part A (addition) and Part B (subtraction) are both polynomials, the polynomials re closed under addition and subtraction.

A) 5x² + 7x - 4

B) 5x³ - x² - 8x + 1

C) Yes, polynomial are closed under addition and subtraction. This means that after subtraction or addition of two polynomials, new polynomial is obtained, as was seen in part A and part B

Step-by-step explanation:

A) total length of the two sides = side 1 + side 2 =  

3x² − 2x − 1

+

2x² + 9x  − 3

5x² + 7x - 4

B) Perimeter = side 1 + side 2 + side 3

side 3 = Perimeter - (side 1 + side 2) =

5x³ + 4x² − x − 3

-

        5x² + 7x - 4

5x³ - x² - 8x + 1