Math 71

1. Math 71 5.1 – Introduction to Polynomials and Polynomial Functions

2. Polynomials A number or a number multiplied by variables is called a ________________ (also called a ________).

3. Polynomials A number or a number multiplied by variables is called a ________________ (also called a ________). monomial

4. Polynomials A number or a number multiplied by variables is called a ________________ (also called a ________). monomial term

5. Polynomials Ex 1.Which of the following are monomials? 17

6. Polynomials Ex 1.Which of the following are monomials? 17

7. Polynomials The sum of the exponents of the variables is called the ________________ of a monomial. The numerical factor in a monomial is called the ___________________.

8. Polynomials The sum of the exponents of the variables is called the ________________ of a monomial. The numerical factor in a monomial is called the ___________________. degree

9. Polynomials The sum of the exponents of the variables is called the ________________ of a monomial. The numerical factor in a monomial is called the ___________________. degree coefficient

10. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

11. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

12. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

13. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

14. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

15. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

16. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

17. Polynomials Ex 2.What are the degrees and coefficients of the following monomials?

18. Polynomials One or more monomials added together is called a ___________________. ex: Thehighest degree of all terms in a polynomial is called the _________________ of the polynomial. ex: What is degree of above polynomial?

19. Polynomials One or more monomials added together is called a ___________________. ex: Thehighest degree of all terms in a polynomial is called the _________________ of the polynomial. ex: What is degree of above polynomial? polynomial

20. Polynomials One or more monomials added together is called a ___________________. ex: Thehighest degree of all terms in a polynomial is called the _________________ of the polynomial. ex: What is degree of above polynomial? polynomial degree

21. Polynomials One or more monomials added together is called a ___________________. ex: Thehighest degree of all terms in a polynomial is called the _________________ of the polynomial. ex: What is degree of above polynomial? polynomial degree

22. Polynomials A polynomial with 2 terms is called a ____________________. A polynomial with 3 terms is called a ____________________.

23. Polynomials A polynomial with 2 terms is called a ____________________. A polynomial with 3 terms is called a ____________________. binomial

24. Polynomials A polynomial with 2 terms is called a ____________________. A polynomial with 3 terms is called a ____________________. binomial trinomial

25. Adding Polynomials Ex 3.Add:

26. Adding Polynomials Ex 3.Add:

27. Subtracting Polynomials Ex 4.Subtract:

28. Subtracting Polynomials Ex 4.Subtract:

29. Subtracting Polynomials Ex 4.Subtract:

30. Polynomial Functions Here’s a polynomial function: And here’s what looks like:

31. Polynomial Functions In general, polynomial functions are __________ (i.e. only have rounded curves with no sharp corners) and ________________ (i.e. don’t have breaks and can be drawn without lifting pencil).

32. Polynomial Functions In general, polynomial functions are __________ (i.e. only have rounded curves with no sharp corners) and ________________ (i.e. don’t have breaks and can be drawn without lifting pencil). smooth

33. Polynomial Functions In general, polynomial functions are __________ (i.e. only have rounded curves with no sharp corners) and ________________ (i.e. don’t have breaks and can be drawn without lifting pencil). smooth continuous

34. Polynomial Functions The following are not the graphs of polynomial functions:

35. Polynomial Functions The following are not the graphs of polynomial functions: Not smooth (Sharp corner—ow!)

36. Polynomial Functions The following are not the graphs of polynomial functions: Not smooth (Sharp corner—ow!) Not continuous at

37. End Behavior The question “What is the end behavior of a function?” means “What does the function do as it goes to the far left or far right?” That is, does it go up? Go down? Wiggle up and down? Get flatter?

38. End Behavior The question “What is the end behavior of a function?” means “What does the function do as it goes to the far left or far right?” That is, does it go up? Go down? Wiggle up and down?

39. End Behavior For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree). Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values…

40. End Behavior For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree). Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values… leading term

41. End Behavior For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree). Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values… leading term highest

42. End Behavior For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree). Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values… leading term highest

43. End Behavior For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree). Why? Let’s look at what happens to the polynomial function when we plug in big positive -values… leading term highest

44. End Behavior 1 -2  10 1897  100 1989997  1000 1998999997 3 The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.

45. End Behavior 1 -2  10 1897  100 1989997  1000 1998999997 3 The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.

46. End Behavior 1 -2  10 1897  100 1989997  1000 1998999997 3 The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.

47. End Behavior 1 -2  10 1897  100 1989997  1000 1998999997 3 The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.

48. End Behavior 1 -2  10 1897  100 1989997  1000 1998999997 3 The term grows faster than the and terms. So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.

49. So, to determine the end behavior of a polynomial function, think about what would happen if you plugged in a really big negative -value and a really big positive -value into the leading term. (In fact, any negative # and positive # will do.)

50. Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________. Graph ________to left and ________to right.