Question

The lengths of three sides of a trapezoid are shown below:

Side 1: 11z2 − 4z + 2
 Side 2: −2z + 3 + 12z2
 Side 3: 3 − 3z + 13z2
  The perimeter of the trapezoid is 5z3 + 40z2 + 7z − 15.
 Part A: What is the total length of sides 1, 2, and 3, of the trapezoid?
Part B: What is the length of the fourth side of the trapezoid?
Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction?

Answer 1

Part A: the total length is 36z^2-9z+8

Part B: the length of the fourth side is 5z^3+4z^2+16z-23

Part C: yes, because when you add polynomials you get a polynomial and when you subtract polynomials you get a polynomial

Answer 2

Given that the lengths of three sides of a trapezoid are shown as follows:
Side 1:
`11z^2 - 4z + 2`
Side 2:
`-2z + 3 + 12z^2`
Side 3:
`3 - 3z + 13z^2`
and The perimeter of the trapezoid is
`5z^3 + 40z^2 + 7z - 15.`

PART A:
The total length of sides 1, 2, and 3 of the trapezoid is obtained by the sum of the polynomials representing the length of sides 1, 2, and 3.

Thus, total length of sides 1, 2, and 3 is given by

`(11z^2-4z + 2)+(-2z + 3 + 12z^2)+(3 - 3z + 13z^2) \\ \\ =11z^2-4z + 2-2z + 3 + 12z^2+3 - 3z + 13z^2 \\ \\ =36z^2-9z+8`


PART B:
The length of the fourth side of the trapezoid is given by the difference between the perimeter of the trapezoid and the length of sides 1, 2, and 3.

Thus, the length of the fourth side of the trapezoid is given by

`(5z^3 + 40z^2 + 7z - 15)-(36z^2-9z+8) \\ \\ =5z^3 + 40z^2 + 7z - 15-36z^2+9z-8 \\ \\ =5z^3+4z^2+16z-23`


PART C:
From part A, we can see that the addition of polynomials results in a polynomial.
Likewise, the subtraction of polynomials results in a polynomial.

Therefore, we can say that the polynomials are closed under addition and subtraction.