PRECALCULUS ADVANCED
PRECALCULUS ADVANCED
WORKSHEET 3 ON ZEROS OF POLYNOMIAL FUNCTIONS
Ex. Given the function [pic] and the synthetic division charts below:
3 8 0 [pic]1 3 8 0 [pic]1
0 3 8 0 [pic]1 0 3 8 0 [pic]1
1 3 11 11 10 [pic]1 3 5 [pic]5 4
2 3 14 28 55 [pic]2 3 2 [pic]4 7
3 3 17 51 152 [pic]3 3 [pic]1 3 [pic]10
4 3 20 80 319 [pic]4 3 [pic]4 16 [pic]65
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
________________________________________________________________________________
Ex. Given the function [pic]and the synthetic division charts
below:
[pic]8 10 37 [pic]50 9 [pic]8 10 37 [pic]50 9
0 [pic]8 10 37 [pic]50 9 0 [pic]8 10 37 [pic]50 9
1 [pic]8 2 39 [pic]11 [pic]2 [pic]1 [pic]8 18 19 [pic]69 78
2 [pic]8 [pic]6 25 0 9 [pic]2 [pic]8 26 [pic]15 [pic]20 49
3 [pic]8 [pic]14 [pic]5 [pic]65 [pic]186 [pic]3 [pic]8 34 [pic]65 145 [pic]426
4 [pic]8 [pic]22 [pic]51 [pic]254 [pic]1007 [pic] 4 [pic]8 42 [pic]131 474 [pic]1887
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
1. Given the function [pic]and the synthetic division charts
below:
[pic]5 7 49 [pic]50 [pic]5 7 49 [pic]50
0 [pic]5 7 49 [pic]50 0 [pic]5 7 49 [pic]50
1 [pic]5 2 51 1 [pic]1 [pic]5 12 37 [pic]87
2 [pic]5 [pic]3 43 36 [pic]2 [pic]5 17 15 [pic]80
3 [pic]5 [pic]8 25 25 [pic]3 [pic]5 22 [pic]17 1
4 [pic]5 [pic]13 [pic]3 [pic]62 [pic]4 [pic]5 27 [pic]59 186
5 [pic]5 [pic]18 [pic]41 [pic]255 [pic]5 [pic]5 32 [pic]111 505
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
________________________________________________________________________________________________
2. Given the function [pic]and the synthetic division charts
below:
1 1 0 [pic]5 1 1 0 [pic]5
0 1 0 1 [pic]5 0 1 0 1 [pic]5
1 1 1 2 [pic]3 [pic]1 1 [pic]1 2 [pic]7
2 1 2 5 5 [pic]2 1 [pic]2 5 [pic]15
3 1 3 10 25 [pic]3 1 [pic]3 10 [pic]35
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
3. Given the function [pic]and the synthetic division charts
below:
1 [pic]1 [pic]10 5 18 1 [pic]1 [pic]10 5 18
0 1 [pic]1 [pic]10 5 18 0 1 [pic]1 [pic]10 5 18
1 1 0 [pic]10 [pic]5 13 [pic]1 1 [pic]2 [pic]8 13 5
2 1 1 [pic]8 [pic]11 [pic]4 [pic]2 1 [pic]3 [pic]4 13 [pic]8
3 1 2 [pic]4 [pic]7 [pic]3 [pic]3 1 [pic]4 2 [pic]1 21
4 1 3 2 13 70 [pic]4 1 [pic]5 10 [pic]35 158
5 1 4 10 55 293 [pic]5 1 [pic]6 20 [pic]95 493
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
______________________________________________________________________________________________
4. Given the function [pic]and the synthetic division charts
below:
1 0 [pic]3 [pic] 6 [pic]2 1 0 [pic]3 [pic]6 [pic]2
0 1 0 [pic]3 [pic]6 [pic]2 0 1 0 [pic]3 [pic]6 [pic]2
1 1 1 [pic]2 [pic]8 [pic]10 [pic]1 1 [pic]1 [pic]2 [pic]4 2
2 1 2 1 [pic]4 [pic]10 [pic]2 1 [pic]2 1 [pic]8 14
3 1 3 6 12 34 [pic]3 1 [pic]3 6 [pic]24 70
4 1 4 13 46 182 [pic]4 1 [pic]4 13 [pic]58 230
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
5. Given the function [pic]and the synthetic division charts
below:
[pic]2 0 3 [pic]20 65 [pic]2 0 3 [pic]20 65
0 [pic]2 0 3 [pic]20 65 0 [pic]2 0 3 [pic]20 65
1 [pic]2 [pic]2 1 [pic] 19 46 [pic]1 [pic]2 2 1 [pic]21 86
2 [pic]2 [pic]4 [pic]5 [pic]30 5 [pic]2 [pic]2 4 [pic]5 [pic]10 85
3 [pic]2 [pic]6 [pic]15 [pic]65 [pic]130 [pic]3 [pic]2 6 [pic]15 25 [pic]10
4 [pic]2 [pic]8 [pic]29 [pic]136 [pic]479 [pic]4 [pic]2 8 [pic]29 96 [pic]319
5 [pic]2 [pic]10 [pic]47 [pic]255 [pic]1210 [pic]5 [pic]2 10 [pic]47 215 [pic]1010
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
_______________________________________________________________________________________________
6. Given the function [pic]and the synthetic division charts
below:
1 [pic]6 10 [pic]6 10 1 [pic]6 10 [pic]6 10
0 1 [pic]6 10 [pic]6 10 0 1 [pic]6 10 [pic]6 10
1 1 [pic]5 5 [pic]1 9 [pic]1 1 [pic]7 17 [pic]23 33
2 1 [pic]4 2 [pic]2 6 [pic]2 1 [pic]8 26 [pic]58 126
3 1 [pic]3 1 [pic]3 1 [pic]3 1 [pic]9 37 [pic]117 361
4 1 [pic]2 2 2 18 [pic]4 1 [pic]10 50 [pic]206 834
(a) Sketch the function on the axes on the right below. Be sure to label the y-axis tick marks.
(b) Locate the zeros of the function between consecutive integers. __________________________
(c) Fill in the chart for your function:
|No. of Positive Zeros |No. of Negative Zeros |No. of Imaginary Zeros |Total No. of Zeros |
| | | | |
(d) Use Descartes’ Rule to predict the number of
positive and negative real zeros, and make a PNI chart.
Possible positive = ____________________
Possible negative = ____________________
(e) Do your answers to (c) and (d) agree? _________
(f) For this function, all of the positive real zeros are
less than or equal to _______. (Name the smallest
positive integer that makes this true.)
(g) For this function, all of the negative real zeros are
greater than or equal to _______. (Name the largest
negative integer that makes this true.)
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