Are polynomials closed under division? Use your answers to Problem 2 to help explain. Then tell why polynomials must be closed under addition, subtraction, and multiplication. Use your answers to Problems 1 and 3 to support your explanation.

Juan has a rectangular corral for his horses. The length of his horse corral is 10 ft longer than 3 times its width. Jeremiah has a rectangular corral for his cattle. The length of his cattle corral is 8 ft longer than 4 times its width. Both corrals have the same width. Let x represent this width, in feet.

1. Write a polynomial, in standard form, for each of the following. Show your work. Classify each polynomial by its degree and by its number of terms.
2. the perimeter of each corral
3. the difference between the perimeters of Jeremiah's and Juan's corrals
4. the area of each corral
5. the sum of the areas of both corrals

Answer:

1. Juan and Jeremiah consider the dimensions of their corrals.
2. Write an expression for the ratio of the width of Juan's corral to the width of Jeremiah's corral. Simplify the expression. Is this expression a polynomial? Explain.
3. Write an expression for the ratio of the length of Juan's corral to the width of his corral. Is this expression a polynomial? Explain.

Answer:

1. Juan plans to add 4 ft to the width of his corral, and then adjust the length so that it is 5 times the new width. Write a polynomial, in standard form, for each of the following. Show your work. Classify each polynomial by its degree and by its number of terms.

(a) the perimeter of Juan's corral after the changes are made

(b) the area of Juan's corral after the changes are made

answer: