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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.

A) The total length of the two sides is

B) The total length of the third sides is

C) Polynomials are closed under addition and subtraction.

Step-by-step explanation:

Here, the sides of the given triangle are:

Side 1:

Side 2:

The perimeter of the triangle  =

(1) The total length of the two sides, 1 and 2

=  Length of ( Side 1 + Side 2)

=

So, the total length of the two sides is

(2) The Perimeter of the Triangle  = Sum of all 3 sides

⇒Length of ( Side 1 + Side 2 + Side 3) =

or, Sum of two sides  + Third side =  

or,   +  Third side =  

⇒Third side  =  

                     =

So, the total length of the third sides is

C) closed under addition and subtraction:

Polynomials are always CLOSED under addition and subtraction,as after adding two polynomials, we always get another polynomial.

Similarly, after subtraction two polynomials, resultant is also a polynomial.

As it is evident from A and B also.

THE BEST AND MOST SIMPLE ANSWER:

Part A: Length of 1+2  

= Setting Problem up: (8x^2-5x-2) + (7x-x^2+3)  

=  Combining like terms: (8x^2+-x^2) + (-5x+7x) + (-2+3)  

= Solving/ 7x^2+2x+1  

Part B: Perimeter = 1+2+3= 4x^3-3x^2+2-6

We already know that 1+2 = 7x^2+2x+1 so all we need to do is subtract  7x^2+2x+1  from  4x^3-3x^2+2-6.

= Setting problem up: (4x^3-3x^2+2x-6)-(7x^2+2x+1)

= Combining like terms:  (4x^3) + (-3x+(-7x)) + (2x + (-2x)) +(-6+(-1))

= Solving/  4x3−10x2−7  

Part C: Yes they do. Polynomials are always closed under addition and subtraction.