1. Warm Up If you add two even numbers do you always get an even number? If you add two odd numbers do you always get an odd number

2. Essential Question:  What is Closure?

6.  MGSE9-12.A.APR.1 Add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations.

7. Section4: Closed Systems

8. Vocabulary : Set: A collection of numbers Element: A number in the set Example: Set- {1,2,3,4,…} Element- 1 is an element of the set above

9. What is closure? Closure: When you combine any two elements of the set, the result is also included in the set.

10. Is this set closed under multiplication {-1, 0, 1} Put this problem in your notes

11. Is this set closed under addition {1,2,0} Put this problem in your notes

12. Show that the set of irrational numbers is not closed under multiplication. Put this problem in your notes

13. Closure Polynomials will be closed under an operation if the operation produces another polynomial.

14. Closure Under Addition (2x 2 + 3x + 4) + (x 2 - 5x - 3) = 3x 2 - 2x + 1 When adding polynomials, the variables and their exponents do not change. Only their coefficients will possibly change. This guarantees that the sum has variables and exponents which are already classified as belonging to polynomials. Polynomials are closed under addition.

15. Closure under Subtraction (2x 2 + 3x + 4) - (x 2 - 5x - 3) = x 2 + 8x + 7 When subtracting polynomials, the variables and their exponents do not change. Only their coefficients will possibly change. This guarantees that the difference has variables and exponents which are already classified as belonging to polynomials. Polynomials are closed under subtraction.

16. Closure Under Multiplication (x + 1)(x 2 + 4x + 3) = x 3 + 5x 2 + 7x + 3 When multiplying polynomials, the variables' exponents are added, according to the rules of exponents. Remember that the exponents in polynomials are whole numbers. The whole numbers are closed under addition, which guarantees that the new exponents will be whole numbers. Consequently, polynomials are closed under multiplication.

17. Are odd numbers closed under multiplication? Try multiplying some odd numbers together Put this problem in your notes

18. Are even numbers closed under multiplication? Put this problem in your notes

19. Are real numbers closed under addition? Put this problem in your notes