The lengths of two sides of a triangle are shown.
Side 1: 3x2 − 2x − 1
Side 2: 9x + 2x2 − 3
The perimeter of the triangle is 5x3 + 4x2 − x − 3.
Part A: What is the total length of the two sides, 1 and 2, of the triangle? Show your work.(4 points)
Part B: What is the length of the third side of the triangle? Show your work. (4 points)
Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer. (2 points)
sum of 2 given sides = 3x^2 − 2x − 1 + 9x + 2x^2 − 3
= 5x^2 + 7x - 4
third side = perimeter - (sum of 2 given sides)
= 5x^3 + 4x^2 − x − 3 - (5x^2 + 7x - 4)
= 5x^3 - x^2 - 8x + 1
i need help with c. i alr have the other parts.
polynomials are closed under addition and subtraction, if the
results after addition and/or subtraction yields a polynomial.
Since this is what happened, it is closed
jesus christ
Part A: To find the total length of the two sides of the triangle, we simply need to add the lengths of the two sides together. Thus, we add Side 1 and Side 2:
(3x^2 - 2x - 1) + (9x + 2x^2 - 3)
Next, we combine like terms by adding the coefficients of the same degree of x:
(3x^2 + 2x^2) + (-2x + 9x) + (-1 - 3)
Simplifying further:
5x^2 + 7x - 4
So, the total length of the two sides of the triangle is 5x^2 + 7x - 4.
Part B: To find the length of the third side of the triangle, we subtract the sum of the lengths of Side 1 and Side 2 from the perimeter of the triangle:
(5x^3 + 4x^2 - x - 3) - (5x^2 + 7x - 4)
Again, we combine like terms:
(5x^3 - 5x^2) + (4x^2 - 7x) + (-x + 4 - 3)
Simplifying further:
5x^3 - x^2 - 7x + 1
Therefore, the length of the third side of the triangle is 5x^3 - x^2 - 7x + 1.
Part C: To determine if the answers for Part A and Part B show that the polynomials are closed under addition and subtraction, we need to check if the sum and difference of polynomials still results in a polynomial.
In Part A, when we added Side 1 and Side 2, we obtained the polynomial 5x^2 + 7x - 4. Since this is a polynomial, it confirms that polynomials are closed under addition.
In Part B, when we subtracted the sum of Side 1 and Side 2 from the perimeter of the triangle, we obtained the polynomial 5x^3 - x^2 - 7x + 1. As this is also a polynomial, it indicates that polynomials are closed under subtraction.
Therefore, the answers for Part A and Part B demonstrate that the polynomials in this case are closed under addition and subtraction.