The lengths of two sides of a triangle are shown.
Side 1: 8x2 − 5x − 2
Side 2: 7x − x2 + 3
The perimeter of the triangle is 4x3 − 3x2 + 2x − 6.
Part A: What is the total length of the two sides, 1 and 2, of the triangle? Show your work.
Part B: What is the length of the third side of the triangle? Show your work.
Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer.
Part A:
To find the total length of the two sides, we simply add them together:
(8x^2 - 5x - 2) + (7x - x^2 + 3)
= -x^2 + 15x + 1
Therefore, the total length of the two sides is -x^2 + 15x + 1.
Part B:
To find the length of the third side, we need to use the fact that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So we set up an inequality:
(8x^2 - 5x - 2) + (7x - x^2 + 3) > (length of third side)
Simplifying, we get:
6x^2 + 2x + 1 > (length of third side)
Now we use the fact that the perimeter of the triangle is equal to the sum of the lengths of all three sides:
(length of side 1) + (length of side 2) + (length of third side) = 4x^3 - 3x^2 + 2x - 6
Substituting in the expressions for side 1 and side 2, we get:
(8x^2 - 5x - 2) + (7x - x^2 + 3) + (length of third side) = 4x^3 - 3x^2 + 2x - 6
Simplifying, we get:
-x^2 + 15x + 1 + (length of third side) = 4x^3 - 3x^2 + 2x - 6
Now we can solve for the length of the third side:
length of third side = 4x^3 - 2x^2 - 13x - 7
Therefore, the length of the third side is 4x^3 - 2x^2 - 13x - 7.
Part C:
To see if the polynomials are closed under addition and subtraction, we need to check whether the sum or difference of any two polynomials in the form given will also be of that form.
For Part A, we found that the total length of the two sides was -x^2 + 15x + 1. To check if this polynomial is of the same form, we compare it to the general form of a polynomial in this problem:
ax^2 + bx + c
We can see that the polynomial -x^2 + 15x + 1 is indeed of this form, with a = -1, b = 15, and c = 1. Therefore, the polynomials are closed under addition.
For Part B, we found that the length of the third side was 4x^3 - 2x^2 - 13x - 7. To check if this polynomial is of the same form, we again compare it to the general form of a polynomial in this problem:
ax^2 + bx + c
We can see that the polynomial 4x^3 - 2x^2 - 13x - 7 is not of this form, since it has a degree of 3 rather than 2. Therefore, the polynomials are not closed under subtraction.
Overall, we can say that the polynomials are closed under addition, but not under subtraction.