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.
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.