Question

The lengths of two sides of a triangle are shown below:

Side 1: 3x^2 − 4x − 1

Side 2: 4x − x^2 + 5

The perimeter of the triangle is 5x^3 − 2x^2 + 3x − 8.

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

Part B: What is the length of the third side of the triangle? (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)

Answer 1

Explanation:

Part A: To find the total length of the two sides, you need to add the lengths of Side 1 and Side 2.

Side 1: 3x^2 − 4x − 1

Side 2: 4x − x^2 + 5

Adding these two expressions together, we get:

(3x^2 − 4x − 1) + (4x − x^2 + 5)

Rearranging the terms, we have:

(3x^2 - x^2) + (-4x + 4x) + (-1 + 5)

Combining like terms, we get:

2x^2 + 4

So, the total length of Side 1 and Side 2 is 2x^2 + 4.

Part B: The length of the third side of the triangle can be found by subtracting the sum of Side 1 and Side 2 from the perimeter of the triangle.

Perimeter of the triangle: 5x^3 − 2x^2 + 3x − 8

Total length of Side 1 and Side 2: 2x^2 + 4

Subtracting the sum of Side 1 and Side 2 from the perimeter, we get:

(5x^3 − 2x^2 + 3x − 8) - (2x^2 + 4)

Expanding and simplifying, we have:

5x^3 − 2x^2 + 3x − 8 - 2x^2 - 4

Combining like terms, we get:

5x^3 - 4x^2 + 3x - 12

So, the length of the third side of the triangle is 5x^3 - 4x^2 + 3x - 12.

Part C: The answers for Part A and Part B do show that the polynomials are closed under addition and subtraction. When we added the lengths of Side 1 and Side 2, we obtained the polynomial expression 2x^2 + 4, which is a polynomial. When we subtracted the sum of Side 1 and Side 2 from the perimeter of the triangle, we obtained the polynomial expression 5x^3 - 4x^2 + 3x - 12, which is also a polynomial. Therefore, both addition and subtraction of the polynomials resulted in valid polynomial expressions, indicating closure under these operations.