The lengths of two sides of a triangle are shown. Side 1: 3x2 − 4x − 1 Side 2: 4x − x2 + 5 The perimeter of the triangle is 5x3 − 2x2 + 3x − 8. 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)

A.   2x^2 + 4.

B    5x^3 - 4x^2 + 3x - 12.

Yes, ( see Part C below).

Step-by-step explanation:

Part A.

3x^2 − 4x − 1 + 4x -  x^2 + 5

= 2x^2 + 4 (answer).

Part B.

Length of 3rd side =  perimeter - length of sides 1 and 2,

=  5x^3 − 2x^2 + 3x −  8 - (2x^2 + 4)

=  5x^3 - 4x^2 + 3x - 12, (answer)

Part C.

Yes,  because  the results of the additions and subtractions are both polynomials, as well as all the sides being polynomials.

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