Question

(PLZ HELP STAT) The lengths of three sides of a trapezoid are shown below:

Side 1: 16z2 − 4z + 2

Side 2: −2z + 21 + 7z2

Side 3: 2 − 3z + 18z2

The perimeter of the trapezoid is 17z3 + 60z2 + 2z − 5.

Part A: What is the total length of sides 1, 2, and 3, of the trapezoid? (4 points)

Part B: What is the length of the fourth side of the trapezoid? (3 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. (3 points)

Answer 1

Hey there :)

The perimeter formula of a trapezium is:
P = a + b + c + d ( adding all sides )

Perimeter is given : P = 17z³ + 60z² + 2z - 5

Part A :

Total lengths of sides 1, 2 and 3 will be just adding the given lengths : 
1 → 16z² - 4z + 2
2 → 7z² - 2z + 21
3 → 18z² - 3z + 2

side 1 + side 2 + side 3
( 16z² - 4z + 2 ) + ( 7z² - 2z + 21 ) + ( 18z² - 3z + 2 )

Combine like terms together:

16z² + 7z² + 18z² - 4z - 2z - 3z + 2 +21 + 2
41z² - 9z + 25

Part B:

Length of 4th side will be the total perimeter minus the total length of sides 1, 2 and 3:

( 17z³ + 60z² + 2z - 5 ) - ( 41z² - 9z + 25 )

Combine like terms

17z³ + 60z² - 41z² + 2z + 9z - 5 - 25
17z³ + 19z² + 11z - 30

Part C:

Addition of polynomials result in polynomials as well as subtraction of polynomials result in polynomials.

Therefore, this polynomial is closed under addition and subtraction.