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 answerv

Let's solve each part of this question step-by-step.

### Part A: Finding the Total Length of the Two Sides (Side 1 and Side 2)
The lengths of the two sides are given by the expressions:
- **Side 1**: \(3x^2 - 2x - 1\)
- **Side 2**: \(9x + 2x^2 - 3\)

To find the total length of these two sides, we add the polynomials for Side 1 and Side 2:

\[
(3x^2 - 2x - 1) + (9x + 2x^2 - 3)
\]

Combine like terms:
- The \(x^2\) terms: \(3x^2 + 2x^2 = 5x^2\)
- The \(x\) terms: \(-2x + 9x = 7x\)
- The constant terms: \(-1 - 3 = -4\)

So, the total length of the two sides is:

\[
5x^2 + 7x - 4
\]

### Part B: Finding the Length of the Third Side
The perimeter of the triangle is given as:

\[
5x^3 + 4x^2 - x - 3
\]

The perimeter is the sum of all three sides. We already have the combined length of Side 1 and Side 2 as \(5x^2 + 7x - 4\). To find the length of the third side, we subtract the combined length of Side 1 and Side 2 from the perimeter:

\[
\text{Third Side} = (5x^3 + 4x^2 - x - 3) - (5x^2 + 7x - 4)
\]

Now, distribute the subtraction and combine like terms:

1. The \(x^3\) term remains: \(5x^3\)
2. The \(x^2\) terms: \(4x^2 - 5x^2 = -x^2\)
3. The \(x\) terms: \(-x - 7x = -8x\)
4. The constant terms: \(-3 + 4 = 1\)

So, the length of the third side is:

\[
5x^3 - x^2 - 8x + 1
\]

### Part C: Closure of Polynomials Under Addition and Subtraction
To determine if polynomials are closed under addition and subtraction, we check if the results of these operations produce another polynomial.

1. In **Part A**, we added two polynomials, \(3x^2 - 2x - 1\) and \(9x + 2x^2 - 3\), resulting in \(5x^2 + 7x - 4\), which is a polynomial.
2. In **Part B**, we subtracted two polynomials, \(5x^3 + 4x^2 - x - 3\) and \(5x^2 + 7x - 4\), resulting in \(5x^3 - x^2 - 8x + 1\), which is also a polynomial.

Since the results of both addition and subtraction are polynomials, this demonstrates that polynomials are closed under addition and subtraction. Therefore, **yes**, polynomials are closed under these operations.

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Would you like further details on any step?

### Related Questions
1. How would the answer change if Side 1 or Side 2 had a term with \(x^3\)?
2. What if the perimeter had an additional constant term? How would that affect the length of the third side?
3. Can the perimeter ever be zero for a non-zero third side?
4. How would you find the side lengths if given different expressions for Side 1 and Side 2?
5. Are there other polynomial operations that demonstrate closure properties, such as multiplication?

### Tip
When adding or subtracting polynomials, always align terms with the same degree for easier calculation.

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.

Explore how to find the lengths of sides in a polynomial-based triangle and examine the closure property under addition and subtraction. This problem provides a step-by-step solution to calculating the perimeter and determining polynomial operations, suitable for Grades 9-10.

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