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.