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The lengths of two sides of a triangle are shown.

Side 1: 3x^2− 2x − 1

Side 2: 9x + 2x^2− 3

The perimeter of the triangle is 5x^3+ 4x^2 − 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)

The answer to your question is below

Step-by-step explanation:

Side 1:  3x² - 2x - 1

Side 2: 9x + 2x² - 3

Perimeter: 5x³ + 4x² - x - 3

a) Length of side 1 and side 2

Length = (3x² - 2x - 1) + (9x + 2x² - 3)

            = 3x² - 2x - 1 + 9x + 2x² - 3

            = (3x² + 2x²) + (-2x + 9x) + (-1 - 3)

            = 5x² + 7x - 4

b) Length of the third side

Third side = Perimeter - side 1 - side 2

Third side = 5x³ + 4x² - x - 3 - (5x² + 7x - 4)

Third side = 5x³ + 4x² - x - 3 - 5x² - 7x + 4

Third side = 5x³ + (4x² - 5x²) + (-x - 7x) + (-3 + 4)

Third side = 5x³ - x² - 8x + 1

c) Yes, in this problem we did both operations (addition and subtraction) and we can notice that the polynomials are closed. Also they follow the closure property the addition or subtraction of two polynomials gives another polynomial.

Step-by-step explanation:

The length of one side of the triangle is 3x² - 2x - 1

The length of the other side of the triangle is 9x + 2x² - 3

Part A) The total length of the two sides, 1 and 2, of the triangle is

3x² - 2x - 1 + 9x + 2x² - 3

Collecting like terms, it becomes

3x² + 2x² + 9x - 2x - 1 - 3

5x² + 7x - 4

Part B

The perimeter of the triangle is

5x³ + 4x² - x - 3

The length of the third side is

5x³ + 4x² - x - 3 - (5x² + 7x - 4)

Expanding the brackets, it becomes

5x³ + 4x² - x - 3 - 5x² - 7x + 4

Collecting like terms, it becomes

5x³ - 5x² + 4x² - 7x - x - 3 + 4

5x³ - x² - 8x + 1

Part C

Yes, the polynomials are closed under addition and subtraction because these operations produced another polynomial.

a

step-by-step explanation: