Answer :
Let's go through the problem step by step.
### Part A: Total Length of Sides 1 and 2 (4 points)
First, we need to find the total length of the two given sides of the triangle. The lengths are given by the polynomials:
- Side 1: [tex]\( 8x^2 - 5x - 2 \)[/tex]
- Side 2: [tex]\( 7x - x^2 + 3 \)[/tex]
To find the total length of these two sides, we add the polynomials representing each side:
[tex]\[ (8x^2 - 5x - 2) + (7x - x^2 + 3) \][/tex]
Now we combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 8x^2 - x^2 = 7x^2 \)[/tex]
- Combine the [tex]\( x \)[/tex] terms: [tex]\( -5x + 7x = 2x \)[/tex]
- Combine the constant terms: [tex]\( -2 + 3 = 1 \)[/tex]
Thus, the total length of sides 1 and 2 is:
[tex]\[ 7x^2 + 2x + 1 \][/tex]
So, the total length of the two sides is:
[tex]\[ \boxed{7x^2 + 2x + 1} \][/tex]
### Part B: Length of the Third Side (4 points)
We are given the perimeter of the triangle, which is:
[tex]\[ 4x^3 - 3x^2 + 2x - 6 \][/tex]
To find the length of the third side, we subtract the total length of sides 1 and 2 from the perimeter:
[tex]\[ (4x^3 - 3x^2 + 2x - 6) - (7x^2 + 2x + 1) \][/tex]
We will distribute the negative sign through the second polynomial and combine like terms:
[tex]\[ 4x^3 - 3x^2 + 2x - 6 - 7x^2 - 2x - 1 \][/tex]
Combine the terms step by step:
- For [tex]\( x^3 \)[/tex]: [tex]\( 4x^3 \)[/tex]
- For [tex]\( x^2 \)[/tex]: [tex]\( -3x^2 - 7x^2 = -10x^2 \)[/tex]
- For [tex]\( x \)[/tex]: [tex]\( 2x - 2x = 0 \)[/tex]
- For the constants: [tex]\( -6 - 1 = -7 \)[/tex]
So, the length of the third side is:
[tex]\[ 4x^3 - 10x^2 - 7 \][/tex]
Thus, the length of the third side is:
[tex]\[ \boxed{4x^3 - 10x^2 - 7} \][/tex]
### Part C: Closure of Polynomials under Addition and Subtraction (2 points)
When we added the two polynomials in Part A, the result was another polynomial. Similarly, when we subtracted the sum of the two polynomials from the perimeter in Part B, the result was also another polynomial.
- Adding two polynomials [tex]\( (8x^2 - 5x - 2) + (7x - x^2 + 3) \)[/tex] resulted in [tex]\( 7x^2 + 2x + 1 \)[/tex], which is a polynomial.
- Subtracting the total length from the perimeter [tex]\( (4x^3 - 3x^2 + 2x - 6) - (7x^2 + 2x + 1) \)[/tex] resulted in [tex]\( 4x^3 - 10x^2 - 7 \)[/tex], which is also a polynomial.
This demonstrates that polynomials are closed under addition and subtraction, meaning that adding or subtracting polynomials always yields another polynomial.
So the answers to Part A and Part B confirm that polynomials are closed under addition and subtraction:
[tex]\[ \boxed{\text{Yes, polynomials are closed under addition and subtraction because the results are also polynomials.}} \][/tex]
### Part A: Total Length of Sides 1 and 2 (4 points)
First, we need to find the total length of the two given sides of the triangle. The lengths are given by the polynomials:
- Side 1: [tex]\( 8x^2 - 5x - 2 \)[/tex]
- Side 2: [tex]\( 7x - x^2 + 3 \)[/tex]
To find the total length of these two sides, we add the polynomials representing each side:
[tex]\[ (8x^2 - 5x - 2) + (7x - x^2 + 3) \][/tex]
Now we combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 8x^2 - x^2 = 7x^2 \)[/tex]
- Combine the [tex]\( x \)[/tex] terms: [tex]\( -5x + 7x = 2x \)[/tex]
- Combine the constant terms: [tex]\( -2 + 3 = 1 \)[/tex]
Thus, the total length of sides 1 and 2 is:
[tex]\[ 7x^2 + 2x + 1 \][/tex]
So, the total length of the two sides is:
[tex]\[ \boxed{7x^2 + 2x + 1} \][/tex]
### Part B: Length of the Third Side (4 points)
We are given the perimeter of the triangle, which is:
[tex]\[ 4x^3 - 3x^2 + 2x - 6 \][/tex]
To find the length of the third side, we subtract the total length of sides 1 and 2 from the perimeter:
[tex]\[ (4x^3 - 3x^2 + 2x - 6) - (7x^2 + 2x + 1) \][/tex]
We will distribute the negative sign through the second polynomial and combine like terms:
[tex]\[ 4x^3 - 3x^2 + 2x - 6 - 7x^2 - 2x - 1 \][/tex]
Combine the terms step by step:
- For [tex]\( x^3 \)[/tex]: [tex]\( 4x^3 \)[/tex]
- For [tex]\( x^2 \)[/tex]: [tex]\( -3x^2 - 7x^2 = -10x^2 \)[/tex]
- For [tex]\( x \)[/tex]: [tex]\( 2x - 2x = 0 \)[/tex]
- For the constants: [tex]\( -6 - 1 = -7 \)[/tex]
So, the length of the third side is:
[tex]\[ 4x^3 - 10x^2 - 7 \][/tex]
Thus, the length of the third side is:
[tex]\[ \boxed{4x^3 - 10x^2 - 7} \][/tex]
### Part C: Closure of Polynomials under Addition and Subtraction (2 points)
When we added the two polynomials in Part A, the result was another polynomial. Similarly, when we subtracted the sum of the two polynomials from the perimeter in Part B, the result was also another polynomial.
- Adding two polynomials [tex]\( (8x^2 - 5x - 2) + (7x - x^2 + 3) \)[/tex] resulted in [tex]\( 7x^2 + 2x + 1 \)[/tex], which is a polynomial.
- Subtracting the total length from the perimeter [tex]\( (4x^3 - 3x^2 + 2x - 6) - (7x^2 + 2x + 1) \)[/tex] resulted in [tex]\( 4x^3 - 10x^2 - 7 \)[/tex], which is also a polynomial.
This demonstrates that polynomials are closed under addition and subtraction, meaning that adding or subtracting polynomials always yields another polynomial.
So the answers to Part A and Part B confirm that polynomials are closed under addition and subtraction:
[tex]\[ \boxed{\text{Yes, polynomials are closed under addition and subtraction because the results are also polynomials.}} \][/tex]
