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. (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)

Question

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. (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)

Vocaloid · Answer

For part A, add sides 1 and 2. Combine like terms and be careful with signs.

For part B, since all three sides of a triangle add up to get the perimeter,

Perimeter = side 1 + side 2 + side 3

Meaning

Side 3 = perimeter - side 1 - side 2

Since you already summed sides 1 and 2 from part A, you can subtract

Perimeter - (answer from part A) to get side 3

Part C) in order for polynomials to be closed, polynomials must add or subtract to get another polynomial as the solution. Look at your answers from part A and B - are they polynomials? (Do they have negative or non-integer exponents?)

Happyfeet180 · Answer

So, part A is 7x^2 + 2x + 1

SmokeyBrown · Answer

@happyfeet180 wrote:

So, part A is 7x^2 + 2x + 1

Well done. That should be correct

Happyfeet180 · Answer

In part b when i add them together do i have to combine like terms in 4x^3 - 3x^2 + 2x - 6?

Happyfeet180 · Answer

hold on im confused

SmokeyBrown · Answer

@happyfeet180 wrote:

In part b when i add them together do i have to combine like terms in 4x^3 - 3x^2 + 2x - 6?

You will be combining like terms again, but not with addition. Since the perimeter is (4x^3 - 3x^2 + 2x - 6), that would be all 3 sides of the triangle added together. To find the third side, take that expression and *subtract* the value of the two sides you already have. In other words, you want to do: (4x^3 - 3x^2 + 2x - 6) MINUS (7x^2 + 2x + 1)

SmokeyBrown · Answer

(Which Vocaloid said already, actually =D)

Happyfeet180 · Answer

i think the answer for part b is 4x^3 - 4x^2 + 4x - 5

Happyfeet180 · Answer

4x^3 + 4x^2 + 4x - 5 ? is that right

Happyfeet180 · Answer

?

SmokeyBrown · Answer

Apologies for the delay, I was out.

@happyfeet180 wrote:

4x^3 + 4x^2 + 4x - 5 ? is that right

It seems like you ADDED the value of the two expressions, but this would not give you the length of the remaining triangle side. Make sure to SUBTRACT the lengths of the first two parts (which is the answer to part A) from the perimeter of the triangle (the value given in part B). The result should be the length of the remaining side. To understand the reasoning, please consider the following: The triangle perimeter, P, is equal to all the side lengths added, a+b+c. If you want to find the length of side c, for example, you could take the other two values, a and b, and subtract them from the total Perimeter. Remember, P = a+b+c, so P - (a+b) is the same as (a+b+b)-(a+b), which gives you c, the remaining side of the triangle. So, again, what you want to do is (4x^3 - 3x^2 + 2x - 6) MINUS (7x^2 + 2x + 1) (Combining like terms with Subtraction, rather than Addition this time)

Happyfeet180 · Answer

so, 4x^3 - 10x^2 - 7

Happyfeet180 · Answer

@smokeybrown

SmokeyBrown · Answer

@happyfeet180 wrote:

so, 4x^3 - 10x^2 - 7

Yup, I think you've got it! Nicely done :)