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