To solve the given problem, let's break it down step-by-step:
Part A: Calculate the total length of the two sides, Side 1 and Side 2, of the triangle.
The lengths of Side 1 and Side 2 are given as:
- Side 1: [tex]\( 3x^2 - 4x - 1 \)[/tex]
- Side 2: [tex]\( 4x - x^2 + 5 \)[/tex]
To find the total length of the two sides, we add these polynomials together:
[tex]\[ (3x^2 - 4x - 1) + (4x - x^2 + 5) \][/tex]
Combine like terms:
[tex]\[ 3x^2 - x^2 + (-4x + 4x) + (-1 + 5) \][/tex]
Simplify:
[tex]\[ (3x^2 - x^2) + (0x) + (4) \][/tex]
[tex]\[ 2x^2 + 4 \][/tex]
So, the total length of the two sides is:
[tex]\[ 2x^2 + 4 \][/tex]
Part B: Calculate the length of the third side of the triangle.
The perimeter of the triangle is given as:
[tex]\[ 5x^3 - 2x^2 + 3x - 8 \][/tex]
The total length of the two sides (from Part A) is:
[tex]\[ 2x^2 + 4 \][/tex]
To find the length of the third side, we subtract the total length of the two sides from the perimeter:
[tex]\[ (5x^3 - 2x^2 + 3x - 8) - (2x^2 + 4) \][/tex]
Distribute the negative sign and combine like terms:
[tex]\[ 5x^3 - 2x^2 + 3x - 8 - 2x^2 - 4 \][/tex]
Combine like terms:
[tex]\[ 5x^3 - (2x^2 + 2x^2) + 3x + (-8 - 4) \][/tex]
Simplify:
[tex]\[ 5x^3 - 4x^2 + 3x - 12 \][/tex]
So, the length of the third side is:
[tex]\[ 5x^3 - 4x^2 + 3x - 12 \][/tex]
Part C: Do the answers from Part A and Part B show that polynomials are closed under addition and subtraction? Justify your answer.
The concept of closure under addition and subtraction for polynomials means that when you add or subtract two polynomials, the result must also be a polynomial.
In Part A, we added two polynomials:
[tex]\[ (3x^2 - 4x - 1) + (4x - x^2 + 5) \][/tex]
This gave us:
[tex]\[ 2x^2 + 4 \][/tex]
which is a polynomial.
In Part B, we subtracted one polynomial from another:
[tex]\[ (5x^3 - 2x^2 + 3x - 8) - (2x^2 + 4) \][/tex]
This gave us:
[tex]\[ 5x^3 - 4x^2 + 3x - 12 \][/tex]
which is also a polynomial.
Since both the addition and subtraction of these polynomials resulted in expressions that are themselves polynomials, we can conclude that the polynomials are closed under addition and subtraction.
Therefore, the answers for Part A and Part B do indeed show that the polynomials are closed under addition and subtraction.
