Sure, let's walk through the solution step-by-step.
### Part A: What is the total length of the two sides, 1 and 2, of the triangle?
1. Identify the polynomials for the lengths of sides 1 and 2:
- Side 1: \(8x^2 - 5x - 2\)
- Side 2: \(7x - x^2 + 3\)
2. Add the polynomials for sides 1 and 2:
[tex]\[
\begin{align*}
\text{Total Length} &= (8x^2 - 5x - 2) + (7x - x^2 + 3) \\
&= 8x^2 - x^2 + (-5x + 7x) + (-2 + 3) \\
&= 7x^2 + 2x + 1
\end{align*}
\][/tex]
Thus, the total length of the two sides is \(7x^2 + 2x + 1\).
### Part B: What is the length of the third side of the triangle?
1. Identify the polynomial for the perimeter of the triangle:
- Perimeter: \(4x^3 - 3x^2 + 2x - 6\)
2. Subtract the total length of sides 1 and 2 from the perimeter to find the length of the third side:
[tex]\[
\begin{align*}
\text{Third Side} &= \text{Perimeter} - \text{Total Length} \\
&= (4x^3 - 3x^2 + 2x - 6) - (7x^2 + 2x + 1) \\
&= 4x^3 - 3x^2 - 7x^2 + 2x - 2x - 6 - 1 \\
&= 4x^3 - 10x^2 - 7
\end{align*}
\][/tex]
Therefore, the length of the third side is \(4x^3 - 10x^2 - 7\).
### Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer.
1. Addition:
- From Part A, we added the polynomials for sides 1 and 2 and obtained \(7x^2 + 2x + 1\). This is a polynomial.
2. Subtraction:
- From Part B, we subtracted the total length from the perimeter and obtained \(4x^3 - 10x^2 - 7\). This is also a polynomial.
Both the addition and subtraction of the given polynomials result in polynomials. Therefore, the operations of addition and subtraction are closed under the set of polynomials. This confirms that polynomials are closed under these operations.
So, the answers to Parts A and B show that the polynomials are indeed closed under addition and subtraction.
