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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.
Part B: What is the length of the third side of the triangle? Show your work.
Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer.

User Mirka
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1 Answer

3 votes

Answer:

Part A:

To find the total length of the two sides of the triangle, we need to add Side 1 and Side 2:

Side 1: 3x² - 4x - 1

Side 2: 4x - x² + 5

Total length: (3x² - 4x - 1) + (4x - x² + 5)

Simplifying and combining like terms, we get:

Total length: -x² + 7x + 4

Therefore, the total length of the two sides of the triangle is -x² + 7x + 4.

Part B:

To find the length of the third side of the triangle, we need to subtract the sum of Side 1 and Side 2 from the perimeter of the triangle:

Length of Side 3 = Perimeter - (Side 1 + Side 2)

Length of Side 3 = (5x³ - 2x² + 3x - 8) - (3x² - 4x - 1 + 4x - x² + 5)

Simplifying and combining like terms, we get:

Length of Side 3 = 2x³ - 2x² + 7x - 12

Therefore, the length of the third side of the triangle is 2x³ - 2x² + 7x - 12.

Part C:

The answers for Part A and Part B show that the polynomials are closed under addition and subtraction. The sum of Side 1 and Side 2 is a polynomial (-x² + 7x + 4), and the difference between the perimeter of the triangle and the sum of Side 1 and Side 2 is also a polynomial (2x³ - 2x² + 7x - 12). In both cases, the result is a polynomial, which means that the set of polynomials is closed under addition and subtraction.

User Dave New
by
7.3k points
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