Let's answer the question step by step:
### Part A: Total Length of Side 1 and Side 2
To find the total length of side 1 and side 2, we need to add the polynomial expressions for each side.
[tex]\[
\text{Side 1: } 3x^2 - 4x - 1
\][/tex]
[tex]\[
\text{Side 2: } 4x - x^2 + 5
\][/tex]
We add these two polynomials together:
[tex]\[
\begin{align*}
(3x^2 - 4x - 1) + (4x - x^2 + 5) &= 3x^2 - 4x - 1 + 4x - x^2 + 5 \\
&= 3x^2 - x^2 - 4x + 4x - 1 + 5 \\
&= 2x^2 + 4
\end{align*}
\][/tex]
Therefore, the total length of sides 1 and 2 is:
[tex]\[
2x^2 + 4
\][/tex]
### Part B: Length of the Third Side
The perimeter of the triangle is given by the polynomial:
[tex]\[
5x^3 - 2x^2 + 3x - 8
\][/tex]
To find the length of the third side, we subtract the sum of side 1 and side 2 from the perimeter:
[tex]\[
\text{Perimeter: } 5x^3 - 2x^2 + 3x - 8
\][/tex]
[tex]\[
\text{Total length of side 1 and side 2: } 2x^2 + 4
\][/tex]
Subtracting the total length of side 1 and side 2 from the perimeter:
[tex]\[
\begin{align*}
(5x^3 - 2x^2 + 3x - 8) - (2x^2 + 4) &= 5x^3 - 2x^2 + 3x - 8 - 2x^2 - 4 \\
&= 5x^3 - 4x^2 + 3x - 12
\end{align*}
\][/tex]
Therefore, the length of the third side is:
[tex]\[
5x^3 - 4x^2 + 3x - 12
\][/tex]
### Part C: Polynomial Closure under Addition and Subtraction
Polynomials are closed under addition and subtraction if the result of adding or subtracting any two polynomials produces another polynomial.
In Part A, we added two polynomial expressions:
[tex]\[
(3x^2 - 4x - 1) + (4x - x^2 + 5) = 2x^2 + 4
\][/tex]
We observed that the result, [tex]\(2x^2 + 4\)[/tex], is itself a polynomial.
In Part B, we subtracted two polynomial expressions:
[tex]\[
(5x^3 - 2x^2 + 3x - 8) - (2x^2 + 4) = 5x^3 - 4x^2 + 3x - 12
\][/tex]
We observed that the result, [tex]\(5x^3 - 4x^2 + 3x - 12\)[/tex], is also a polynomial.
Thus, the operations conducted in Parts A and B show that the polynomials are closed under addition and subtraction, as the results of such operations were also polynomials.
