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i just need help figuring this out
i dont know, its part of the question The lengths of two sides of a triangle are shown below: 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? (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) but i just need help with that part
can you help?
can you help
Part A: What is the total length of the two sides, 1 and 2, of the triangle? to do that , add the two sides
to add , write down the length of 1 side, put in a "+" sign, and the write the other side
side 1 is \( 3x^2 − 4x − 1 \) side 2 is \( 4x − x^2 + 5 \)
yes. notice the "add part" is simple. you write \[ 3x^2 − 4x − 1 + 4x − x^2 + 5 \] the harder part is combining "like terms" and making it look simple
next part. perimeter is the sum of all 3 sides to find the 3rd side, it is what is left over after subtracting off the two other sides from the total length (i.e. from the perimeter)
wait so part a is just 2x^2+4 ?
in other words: perimeter - (two sides) 5x3 − 2x2 + 3x − 8 - (2x^2+4)
2x^2 + 4 is the sum of the first 2 sides when we add the 3rd side we get the perimeter 2x^2+4 + (3rd side) = perimeter or "solving for (3rd side) (3rd side) = perimeter - (2x^2+4)
most excellent. the tricky part with subtraction is to remember the minus applies to all of the terms in (2x^2+4)
So, is that part B?
the last part, just say if you start with polynomials and add or subtract them, the answer is a polynomial... so polynomials are closed under addition and subtraction
your answer for B is good
i dont understand part C
polynomials are closed under addition and subtraction means: if you add two (or subtract) two polynomials, the answer will be a polynomial sometimes operations are closed... sometimes not. for example. integers are closed under add/subtract integer + integer will always give an integer answer. on the other hand, integers are not closed under division example 4 / 2 is 2 so that is ok, but we could also do 2/4 = 1/2 and that is not an integer. so not closed.
so your saying it will have 2 or more terms?
when we talk about polynomials we have to allow for 0 ,1 or more terms (which contradicts the "poly" (Greek for many) definition ), but people want polynomials closed under subtraction, and if we did not allow 0 then x^2 + x - (x^2+x) =0 would have two polynomials giving a non-polynomial answer. Solution: let 0 be part of the polynomials. any way, the answer to part C is, both adding and subtracting polynomials gave a polynomial as an answer.
do x^2 + x - (x^2+x) =0 and that will be the answer?
for part c
no. (that was an extra example) for part C, say that the answers to part A (adding) and part B (subtracting) both started with polynomials, and had a polynomial as an answer.
showing that polynomials are closed under subtraction/addition