Real Zeros, Factors, and Graphs of Polynomial Functions — Transcript

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1. 0:00Welcome to a lesson on the zeros factors  and graphs of polynomial functions.
2. 0:05In this lesson we'll understand  the connections among the
3. 0:08real rational zeros the factors  and the graphs of polynomial
4. 0:12functions. To begin let's assume we  have a polynomial function of degree n
5. 0:20given in this form here and c is a real  number such that p of c equals zero.
6. 0:26So if we sub the value of c into the function the  function value is zero so we say the real number c
7. 0:33is a zero or root of the polynomial function p of  x it's also true that x equals c is a solution to
8. 0:41the equation p of x equals zero so if we take this  polynomial function and replace p of x with zero
9. 0:48forming an equation c would be a solution to that  equation the quantity x minus c is a factor of the
10. 0:56polynomial function which means it would divide  evenly into the polynomial function and then
11. 1:02finally the point c comma zero is an x intercept  of the graph of the polynomial function p of x
12. 1:10to better understand all of this  let's take a look at some examples
13. 1:15a degree four polynomial function has zeros or  roots of multiplicity one at x equals zero when x
14. 1:23equals three and a zero of multiplicity two  at x equals negative one and has a leading
15. 1:30coefficient of negative two we're first asked to  find the x intercepts of the polynomial function
16. 1:37the x intercepts will occur at the real zeros of  the polynomial function and since we're given the
17. 1:45zeros are x equals zero three and negative  one these would be the x intercepts
18. 1:52so if we give them as ordered pairs if the  x intercept is zero that would be the origin
19. 1:57with coordinates zero comma zero if the zero is  three the x intercept would be three comma zero
20. 2:05and if the zero is negative one the x  intercept would be negative one comma zero
21. 2:11next we're asked to write the  polynomial function in factored form
22. 2:15where the r's would be the zeros of the polynomial  function and the ms would be the multiplicity
23. 2:22so the multiplicity is 1 we'd only have one factor  containing the 0 and if the multiplicity is 2 we'd
24. 2:29actually have 2 factors containing the 0.  so for our polynomial function we'd have p
25. 2:35of x equals notice in this form a would be the  leading coefficient which we know is negative two
26. 2:42so a is negative two and then we have two zeros  with multiplicity one one of them is zero and
27. 2:49one of them is three so one factor would have to  be x minus zero we only have one factor of this
28. 2:56because the multiplicity is one we also have one  factor of x minus three but because negative one
29. 3:04has a multiplicity of two we'd actually have two  factors of x minus negative one or x plus one
30. 3:11so this would be squared notice if we sub  any of these x values into this polynomial
31. 3:17function it would make one of these factors  equal to zero and therefore the polynomial
32. 3:22function would be equal to zero let's go ahead  and clean this up a little bit this is p of x
33. 3:29equals of course x minus zero is just  x so we can write negative two x times
34. 3:36x minus three times x plus one squared and now  we're asked to graph the polynomial function
35. 3:44and because we're given three real  zeros of the polynomial function
36. 3:47our polynomial function will have three x  intercepts but whenever the multiplicity is odd
37. 3:54the graph will cross the x axis but if it's  even the graph will touch the x axis at that
38. 4:00point and then turn in the other direction so  if we take a look at the graph of our function
39. 4:06notice how at x equals negative one we have an x  intercept but it does not cross the x axis it just
40. 4:13touches it and then turns direction and that's  because the multiplicity of the zero x equals
41. 4:18negative one is two or even but then notice that  x equals zero here and at x equals three here the
42. 4:28graph does cross the x axis at those two points  because the multiplicity is odd let's take a look
43. 4:34at another example here we're given the graph  of a degree three polynomial function and we're
44. 4:40first asked to find the x intercepts well the  graph crosses the x axis here at negative two
45. 4:48and it also touches the x axis here  at positive one so those would be the
46. 4:52x intercepts so the coordinates would be  negative two comma zero and one comma zero
47. 5:01and now we're asked to find the real zeros or  real roots of the function and we'll also give the
48. 5:07multiplicity well if there's an x intercept at x  equals negative two that would be one of the zeros
49. 5:16and because it crosses the x axis  here the multiplicity must be odd
50. 5:20and therefore the multiplicity would be one here
51. 5:24it couldn't be three because there's  another x intercept over here
52. 5:30another zero would be x equals one and because it  does not cross the x axis just touches it and then
53. 5:37churns this has an even multiplicity and because  we have a degree three polynomial that means the
54. 5:43multiplicity would be two and now we're asked to  find the equation of the polynomial function let's
55. 5:49set this up over here on the side we would have p  of x here we're not given the leading coefficient
56. 5:55so let's go ahead and just leave it as a times if  x equals negative two is a zero then x plus two
57. 6:03would have to be a factor it has multiplicity one  so we only have one factor of x plus two but then
58. 6:10the zero of x equals one has multiplicity two  so we'd have to have two factors of x minus one
59. 6:17and now to find the value of a though we'll have  to use one of the points given on the function
60. 6:22let's go ahead and use this point here where  the coordinates would be two comma negative two
61. 6:30which means that p of two equals negative two  so now what we're going to do is substitute
62. 6:38two for x and set this equal to  the function value of negative two
63. 6:43and that'll allow us to find a so we would  have a times two plus two that'd be four
64. 6:51times two minus one squared that'd be one squared  equals negative two so this would just be four a
65. 7:00equals negative two divide both sides by  four we have a equals negative one half
66. 7:08which means our polynomial function would be  p of x equals a which is negative one half
67. 7:16times the quantity x plus two times  the quantity x minus one squared
68. 7:24i think we have time for one more example here  we're actually given a polynomial equation and
69. 7:30the first question is what polynomial  function would you graph to verify
70. 7:34your solutions this equation would be equivalent  to the polynomial function where p of x
71. 7:41equals zero so we'll substitute p of x  for 0 to form the polynomial function so
72. 7:48our polynomial function we use to verify  our solutions would just be p of x equals
73. 7:540.25 x the quantity x minus one times the quantity  x minus three times the quantity x plus four times
74. 8:03the quantity x plus two squared next it says  where would you look to verify the solutions
75. 8:10the solutions would be the x intercepts next we're  asked what are the x intercepts of the function
76. 8:18well the x intercepts would be the x  values where the function value would be 0
77. 8:23or also would be the solution to this equation  here and we can see this would equal zero when x
78. 8:28equals one three negative four and negative  two which would be the x intercepts
79. 8:37so we'd have the point one zero  we'd have the point three zero
80. 8:42we'd have the point negative four zero  and we'd have the point negative two zero
81. 8:51and the last question what are the  zeros of the polynomial function
82. 8:54well these would be the same as the x intercepts  except we will also include the multiplicity
83. 9:00so the zeros are x equals one x equals three  and x equals negative four with multiplicity one
84. 9:10and then we have the zero x equals negative  two with multiplicity two because notice how
85. 9:16we have two factors of x plus two let's go  ahead and verify all this by looking at the
86. 9:22graph of our polynomial function again notice  how we have one two three four x intercepts
87. 9:30three of which cross through the x axis and  one that touches and turns so the zeros of
88. 9:35negative four one and three have multiplicity  one and the zero of x equals negative two
89. 9:42has an even multiplicity or a multiplicity of two  so the x intercepts these zeros and the solutions
90. 9:52are all very closely related  i hope you found this helpful