Question

The lengths of two sides of a triangle are shown below:

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? (4 points)

Part B: What is the length of the third side of the triangle? (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)
Can someone explain to me how to do this?

Answer 1

What do you get when you add side1 to side2?

Answer 2

7x^2 + 2x +1 ?

Answer 3

correct

Answer 4

that's part A

Answer 5

since that says "Part A: What is the total length of the two sides, 1 and 2, of the triangle?"

Answer 6

Part B: What is the length of the third side of the triangle?

Answer 7

How do I find out the third side? does the perimeter do something? I think that the perimeter is all sides added, so does that mean I have to subtract the perimeter from part a?

Answer 8

Well to get the perimeter of the triangle, you add the 3 sides

Answer 9

Perimeter = side1 + side2 + side3

Answer 10

So yes, but you subtract in the other order

Perimeter = side1 + side2 + side3

Perimeter - (side1 + side2) = side3

Answer 11

4x^3 -10x^2 - 7 ?

Answer 12

Perimeter - (side1 + side2) = side3

(4x^3 - 3x^2 + 2x - 6) - (7x^2 + 2x +1) = side3

4x^3 - 3x^2 + 2x - 6 - 7x^2 - 2x -1 = side3

4x^3 -10x^2 - 7 = side3

side3 = 4x^3 -10x^2 - 7

so you are correct

Answer 13

Thank you, Do you know what I need to do to find out part c?

Answer 14

Do you know what they mean when they say "closed" in part C?

it's a very specific term when it comes to algebra

Answer 15

no I do not, thats why I am confused.

Answer 16

when they say "closed" they mean that adding two polynomials gives you a polynomial

subtracting two polynomials gives you a polynomial

In general, if an operation is closed, then you won't get some other thing that's not a polynomial

Answer 17

example with integers

adding two integers ALWAYS results in another integer

2+5 = 7
8+3 = 11
9+17 = 26
etc etc

pick any two integers, add them, the result is an integer. So adding integers is closed (you won't add 2 integers to get a non-integer, it's just not possible)

Answer 18

so mine is closed because I got a 3 terms?

Answer 19

so it's like an exclusive club that no one else can join

once you have your members set (eg: integers), you can't go outside the club to get non-integers

Answer 20

it's closed because you got a polynomial as a result

Answer 21

a polynomial is simply the sum of a bunch of monomials

Answer 22

so it is always closed if I am adding or subtracting integers

Answer 23

integers was an example

Answer 24

I mean polynomials?

Answer 25

yes, adding or subtracting polynomials gives you a polynomial

Answer 26

The basic template is

Polynomial1 + Polynomial2 = Polynomial3

Polynomial1 - Polynomial2 = Polynomial3

Answer 27

It is impossible to add two polynomials and get a non-polynomial

Answer 28

Thank you for your help

Answer 29

you're welcome

Answer 30

@jim_thompson5910 so for part c it is closed correct?

Answer 31

That is correct. Adding ANY two polynomials results in some other polynomial. Therefore, polynomial addition is a closed operation. The same can be said about polynomial subtraction.