1 / 76

POLYNOMIALS

POLYNOMIALS. And polynomial functions. Properties of Exponents Evaluating and Graphing Polynomial Functions Polynomial Operations Factoring and Solving Polynomials Remainder and Factor Theorem Rational Roots Fundamental Theorem of Algebra Analyzing Graphs of Polynomial Functions

aliza
Download Presentation

POLYNOMIALS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. POLYNOMIALS And polynomial functions Properties of Exponents Evaluating and Graphing Polynomial Functions Polynomial Operations Factoring and Solving Polynomials Remainder and Factor Theorem Rational Roots Fundamental Theorem of Algebra Analyzing Graphs of Polynomial Functions Modeling With Polynomial Functions All Methods APPENDIX

  2. POLYNOMIALS EXPONENTS And polynomial functions Algebra I Exponent Notes MENU APPENDIX

  3. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Topics: Polynomial vocabulary, direct and synthetic substitution, describing end behavior, sketching polynomial graphs APPENDIX MENU

  4. POLYNOMIALS Evaluating and Graphing (basics) Coefficients: the numbers in front of the variables Hey, what’s the coefficient on the x2 term? Here is an example: And polynomial functions Vocabulary: Polynomial Function: any function like this 1 APPENDIX MENU

  5. POLYNOMIALS Evaluating and Graphing (basics) The number on the end is a coefficient on x0 called theconstant term Here is an example: And polynomial functions Vocabulary: Polynomial Function: any function like this APPENDIX MENU

  6. POLYNOMIALS Evaluating and Graphing (basics) Lead Coefficient Here is an example: And polynomial functions Vocabulary: Polynomial Function: any function like this APPENDIX MENU

  7. POLYNOMIALS Evaluating and Graphing (basics) Degree: The highest power of x in the polynomial Here is an example: And polynomial functions Vocabulary: Polynomial Function: any function like this APPENDIX MENU

  8. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Vocabulary: Zero degree Looks like: Highest power of X: 0 # of potential x-intercepts(roots): 0 (or if y = 0 infinite) Type: Constant APPENDIX MENU

  9. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Vocabulary: First degree Looks like: Highest power of X: 1 # of potential x-intercepts(roots): 1 Type: Linear APPENDIX MENU

  10. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Vocabulary: Second degree Looks like: Highest power of X: 2 # of potential x-intercepts(roots): 2 Type: Quadratic APPENDIX MENU

  11. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Vocabulary: Third degree Looks like: Highest power of X: 3 # of potential x-intercepts(roots): 3 Type: Cubic APPENDIX MENU

  12. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Vocabulary: Fourth degree Looks like: Highest power of X: 4 # of potential x-intercepts(roots): 4 Type: Quartic APPENDIX MENU

  13. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Vocabulary: Fifth degree Looks like: Highest power of X: 5 # of potential x-intercepts(roots): 5 Type: Quintic APPENDIX MENU

  14. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Usually, when we have a polynomial we want to EVALUATE it. What is the value of this polynomial when x is negative three? APPENDIX MENU

  15. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Substitution: Here is a fourth degree polynomial. Evaluate it for x = -3 There are other ways to evaluate a polynomial for a value besides substitution HINT: it is easy to type the function into your calculator and use the table to evaluate APPENDIX MENU

  16. POLYNOMIALS Evaluating and Graphing (basics) x - 3 x - 3 x - 3 x - 3 And polynomial functions Synthetic Substitution: This is another way to put a number in for x. It is faster than the long way, but slower than the calculator shortcut, but we do it to set up something else later APPENDIX MENU

  17. POLYNOMIALS Evaluating and Graphing (basics) x 2 x 2 x 2 x 2 And polynomial functions Synthetic Substitution: Try: APPENDIX MENU

  18. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Synthetic Substitution: APPENDIX MENU

  19. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions Synthetic Substitution: Why did that work? Called nested form APPENDIX MENU

  20. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions When we discuss the graphs of polynomials, we want to know 4 things… …How many times does it changes directions? …where does it cross the x-axis? ..what are the high and low points? …does it go up or down on the ends? …does it go up or down on the ends? APPENDIX MENU

  21. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions DOWN On the left The graph goes ________ On the right The graph goes ______ UP APPENDIX MENU

  22. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions On the left, the graph goes _______ On the right, the graph goes ______ UP DOWN APPENDIX MENU

  23. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions APPENDIX MENU

  24. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions End Behavior: Here are some graphs of some polynomials: APPENDIX MENU

  25. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions End Behavior: We want to know how this graph starts and ends Is the leading coefficient Positive or Negative? Is the degree Even or Odd? We can tell this from the leading term. Ends: Negative Starts: Negative APPENDIX MENU

  26. POLYNOMIALS Evaluating and Graphing (basics) And polynomial functions End Behavior: Here are some graphs of some polynomials: Coefficient: + Degree: Odd Coefficient: - Degree: Odd Starts negative Ends positive Starts positive Ends negative Coefficient: + Degree: Even Coefficient: - Degree: Even Starts positive Ends positive Starts negative Ends negative APPENDIX MENU

  27. POLYNOMIALS Evaluating and Graphing (basics) EQUATION GRAPH Lead Coefficient Degree Starts Ends And polynomial functions End Behavior: + EVEN POSITIVE POSITIVE + ODD NEGATIVE POSITIVE - EVEN NEGATIVE NEGATIVE - ODD POSITIVE NEGATIVE an>0 n is even f(x)→∞ as f(x)→∞ as x→-∞ x→∞ an>0 n is odd f(x)→-∞ as f(x)→∞ as x→-∞ x→∞ an<0 n is even f(x)→-∞ as f(x)→-∞ as x→-∞ x→∞ an<0 n is odd f(x)→∞ as f(x)→-∞ as x→-∞ x→∞ APPENDIX MENU

  28. POLYNOMIALS Evaluating and Graphing (basics) EQUATION GRAPH Lead Coefficient Degree Starts Ends And polynomial functions End Behavior: + EVEN + ODD - EVEN - ODD + EVEN f(x)→∞ as f(x)→∞ as x→-∞ x→∞ + ODD f(x)→-∞ as f(x)→∞ as x→-∞ x→∞ - EVEN f(x)→-∞ as f(x)→-∞ as x→-∞ x→∞ - ODD f(x)→∞ as f(x)→-∞ as x→-∞ x→∞ APPENDIX MENU

  29. POLYNOMIALS Polynomial Operations And polynomial functions APPENDIX MENU

  30. POLYNOMIALS Polynomial Operations And polynomial functions Addition and Subtraction: The first method is called the horizontal method. You learned this as adding like terms. APPENDIX MENU

  31. POLYNOMIALS Polynomial Operations And polynomial functions Addition and Subtraction: The second method is called the vertical method. APPENDIX MENU

  32. POLYNOMIALS Polynomial Operations And polynomial functions Addition and Subtraction: Subtraction APPENDIX MENU

  33. POLYNOMIALS Polynomial Operations And polynomial functions Addition and Subtraction: Vertical subtraction APPENDIX MENU

  34. POLYNOMIALS Polynomial Operations And polynomial functions Addition and Subtraction: MULTIPLICATION You first learned to do this as FOIL, but you really just distribute APPENDIX MENU

  35. POLYNOMIALS Polynomial Operations And polynomial functions Addition and Subtraction: MULTIPLICATION You first learned to do this as FOIL, but you really just distribute Basically, multiply every term in one set of parenthesis by every term in the other APPENDIX MENU

  36. POLYNOMIALS Factoring and Solving And polynomial functions Factoring: Last semester, we factored quadratics… APPENDIX MENU

  37. POLYNOMIALS Factoring and Solving And polynomial functions Factoring: This semester we are going to factor tougher problems APPENDIX MENU

  38. POLYNOMIALS Factoring and Solving And polynomial functions The first way to factor (always check this first): Factor using the greatest common factor APPENDIX MENU

  39. POLYNOMIALS Factoring and Solving And polynomial functions APPENDIX MENU

  40. POLYNOMIALS Factoring and Solving And polynomial functions APPENDIX MENU

  41. POLYNOMIALS Factoring and Solving And polynomial functions APPENDIX MENU

  42. POLYNOMIALS Factoring and Solving And polynomial functions Completely factor the following: APPENDIX MENU

  43. POLYNOMIALS Factoring and Solving And polynomial functions The third factoring method you should check for is pattern factoring Difference of Squares: APPENDIX MENU

  44. POLYNOMIALS Factoring and Solving And polynomial functions Difference of Squares: APPENDIX MENU

  45. POLYNOMIALS Factoring and Solving And polynomial functions New Pattern: Difference of Cubes APPENDIX MENU

  46. POLYNOMIALS Factoring and Solving And polynomial functions New Pattern: Sum of Cubes APPENDIX MENU

  47. POLYNOMIALS Factoring and Solving And polynomial functions FACTOR by GROUPING: We only try this when there are 4 terms. APPENDIX MENU

  48. POLYNOMIALS Factoring and Solving And polynomial functions New Pattern: Difference of Cubes Where’d this come from? Sum of Cubes Did you think it should be 2x2? Don’t just square the x, square the 2! APPENDIX MENU

  49. POLYNOMIALS Factoring and Solving And polynomial functions FACTOR BY GROUPING We try this only if there are 4 terms Take the greatest common factor out of the first 2 terms Take the greatest common factor out of the second 2 terms If these are the same, factor them out APPENDIX MENU

  50. POLYNOMIALS Analyzing Graphs And polynomial functions APPENDIX MENU

More Related