1. Roller coaster polynomials  https://www.youtube.com/watch?v=fmZ8jDCVIwc https://www.youtube.com/watch?v=fmZ8jDCVIwc

2. Polynomials Sec 9.1.1

3. Learning Targets  Vocabulary  Operations between polynomials  Introduction to graphs of polynomials

4. Definitions  Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it means “many terms”  Term: A number, a variable, or the product/quotient of numbers/variables.

5. Polynomial

6. A Term has 3 Components: Coefficient: can be any real number… including zero. Variable Exponent: Can only be positive integers: 0,1,2, 3, These components are very important!!!

7. NOT ALLOWED

8. Check In

9. Naming a Polynomial  We can classify a polynomial based on how many terms it has: Polynomial 7 5x + 2 4x 2 + 3x - 4 6x 3 - 18 # Terms 1 2 3 2 # Terms Name monomial binomial trinomial binomial

10. Naming Cont.  Quadrinomial (4 term) and quintinomial (5 term) also exist, but those names are not often used.  Polynomials Can Have Lots and Lots of Terms  Polynomials can have as many terms as needed, but not an infinite number of terms. infinite For more than 3 terms say: “a polynomial with n terms” or “an n- term polynomial” 11x 8 + x 5 + x 4 - 3x 3 + 5x 2 - 3 “a polynomial with 6 terms” – or – “a 6- term polynomial”

11. Degree of a Term The degree of a term is determined by the exponent of the variable. Term 3 4x -5x 2 18x 5 Degree of Term 0 1 2 5

12. Naming a Polynomial  We can also classify a polynomial based on its highest degree: Polynomial 7 5x + 2 4x 2 + 3x - 4 6x 3 - 18 Degree 0 1 2 3 # Degree Name Constant Linear Quadratic Cubic

13. Putting it All Together Name cubic monomial quadratic monomial constant monomial linear binomial cubic trinomial quadratic trinomial 4 th degree binomial Polynomial -14x 3 -1.2x 2 7x - 2 3x 3 + 2x - 8 2x 2 - 4x + 8 x 4 + 3

14. Standard Form of a Polynomial A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree.

15. Not Standard 6x + 3x 2 - 2 15 - 3x - x+ 5x 4 x + 10 + x 1 + x 2 + x + x 3 Standard 3x 2 + 6x - 2 5x 4 - 4x + 15 2x + 10 x 3 + x 2 + x + 1

16. Finding the Leading Terms Degree

17. Fill in the following table: Leading CoefficientDegreeEnd Behaviors Positive Negative Positive Negative Degree Even Odd End Behavior

18. Leading CoefficientDegreeEnd Behaviors Positive Negative Positive Negative End BehaviorDegree Even Odd

19. Types of Roots  Polynomial solutions are made up of complex roots  A root is where the polynomial’s graph will intersect with the x-axis  A complex root describes two different types of roots:  Real Roots  Imaginary Roots (we will get to these next week)

20. Root Classifications  We classify the type of Real Root based on the degrees of each term and how it interacts with the x-axis.  Types:  Single Root  Double Root  Triple Root  And so on…

21. Examples:  Single Roots

22. Examples:  Double Roots

23. Examples:  Triple Roots

24. You Try  Classify each type of root:

25. Desmos  More on degree and type of roots  https://www.desmos.com/calculator/zj7vtcpmpw https://www.desmos.com/calculator/zj7vtcpmpw

26. Practice

27. #1

28. #2

29. #3

30. Turns In a Graph  What determines the number of turns the graph of a polynomial will have?  End Behavior  Degree of the Leading Term  Degrees of each factor, or the types of roots  The maximum number of turns a polynomial can have is (n-1) where n is the degree of the leading term

31.  http://www.wired.com/images_blogs/wiredscience/2 013/10/600-grey-goblin.gif http://www.wired.com/images_blogs/wiredscience/2 013/10/600-grey-goblin.gif