The lengths of two sides of a triangle are shown. Side 1: 3x2 − 4x − 1 Side 2: 4x − x2 + 5 The perimeter of the triangle is 5x3 − 2x2 + 3x − 8. 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)

A.   2x^2 + 4.

B    5x^3 - 4x^2 + 3x - 12.

Yes, ( see Part C below).

Step-by-step explanation:

Part A.

3x^2 − 4x − 1 + 4x -  x^2 + 5

= 2x^2 + 4 (answer).

Part B.

Length of 3rd side =  perimeter - length of sides 1 and 2,

=  5x^3 − 2x^2 + 3x −  8 - (2x^2 + 4)

=  5x^3 - 4x^2 + 3x - 12, (answer)

Part C.

Yes,  because  the results of the additions and subtractions are both polynomials, as well as all the sides being polynomials.

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1

600,000,000,000, or 6e14

step-by-step explanation: