Algebra II Pre-AP
Notes- More about polynomials
Name: _________________________
Date: ___________________Per ____
1. Given the polynomial and the synthetic division table answer the following questions: P  x   x 4  x3  18x 2  16 x  32
1
1
-18
-16
32
5
1
6
12
44
252
4
1
5
2
-8
0
3
1
4
-6
-34
-70
2
1
3
-12
-40
-48
1
1
2
-16
-32
0
0
1
1
-18
-16
32
-1
1
0
-18
2
30
-2
1
-1
-16
16
0
-3
1
-2
-12
20
-28
-4
1
-3
-6
8
0
-5
1
-4
2
-26
162
a) What are the roots of the polynomial and how can you tell?
b) Find P(3) and P(-1).
2. Given the synthetic division table answer the following questions:
a) What is the polynomial being divided?
b) What is the y-intercept?
c) What are the Possible Rational Roots?
d) There is a root at x = ______.
e) How do we know that there is also a real root between 0 and 1?
f) Could the root between 0 and 1 be rational? Explain.
g) Find all three roots.
-2
-11
-12
9
5
-2
-21
-117
-576
4
-2
-19
-88
-343
3
-2
-17
-63
-180
2
-2
-15
-42
-75
1
-2
-13
-25
-16
0
-2
-11
-12
9
-1
-2
-9
-3
12
-2
-2
-7
2
5
-3
-2
-5
3
0
-4
-2
-3
0
9
-5
-2
-1
-7
44
-6
-2
1
-18
117
-7
-2
3
-33
240
h) Sketch the polynomial
3. Given the synthetic division table answer the following questions:
1
-4
-2
12
a) What is the polynomial being divided? _____________________________________
5
1
1
3
27
b) There is a root at x  _______ The y intercept is __________________
4
1
0
-2
4
3
1
-1
-5
-3
2
1
-2
-6
0
1
1
-3
-5
7
0
1
-4
-2
12
-1
1
-5
3
9
-2
1
-6
10
-8
-3
1
-7
19
-45
-4
1
-8
30
-108
-5
1
-9
43
-203
c) P  3  ________
P  1  ________
P  5  ________
d) The Possible Rational Roots are ____________________________________________
e)
P
N
I
f) A positive real root lies between _______ and _______
g) A negative real root lies between _______ and _______
h) Will the roots in part f and g be rational or irrational? Explain.
i) Are there any imaginary roots? Explain.
4. Given the synthetic division table answer the following questions:
a) What is the polynomial being divided?
b) There is a root at x  _______
c) The Possible Rational Roots are:
d)
P
N
I
2
-7
2
14
-12
-4
2
-15
62
-234
924
-3
2
-13
41
-109
315
-2
2
-11
24
-34
56
-1
2
-9
11
3
-15
0
2
-7
2
14
-12
1
2
-5
-3
11
-1
2
2
-3
-4
6
0
3
2
-1
-1
11
21
4
2
1
6
38
140
e) A real root lies between _______ and _______
f) Find all four roots. (to help us find the roots we can use something called the Least Upper Bound and Greatest Lower Bound, refer to other note sheet)
LUB = ______
GLB = ______
Algebra II Pre AP
Analyze This! Worksheet
Name_______________________________
Date___________________ Period_______
1.
-4
-3
-2
1
0
1
2
3
a.
b.
c.
d.
e.
1
1
1
1
1
1
1
1
1
0 0
3 -13
-4 16 -61 231
-3 9 -24 59
-2 4 -5
-3
-1 1
2 -15
0 0
3 -13
1 1
4
-9
2 4 11
9
3 9 30 77
1,  13
What is the degree of the polynomial?
Locate the roots between consecutive integers.
Are the roots in part b rational, irrational or imaginary? Explain.
Sketch the graph.
Based on the synthetic division chart, what is the upper and lower integer bound for the roots?
2.
-2
-1
0
1
2
a.
b.
c.
d.
e.
possible rational roots
P N I
1 1 2
3 -5
0
2 -7
3 -11 22 -42 77
3 -8 8 -6 -1
3 -5 0
2 -7
3 -2 -2
0 -7
3
1 2
6 5
Make a PNI chart.
List the possible rational roots.
Locate roots between consecutive integers.
Based on the synthetic division chart, what is the upper and lower integer bound for the roots?
From your list in part b, what are the only numbers that need to be considered possible roots?
3. Use the synthetic division chart to answer the following questions.
-5
-4
-3
-2
-1
0
1
2
3
4
4
4
4
4
4
4
4
4
4
14
-6
-2
2
6
10
14
18
22
26
10
-3
40 -203
18 -75
4 -15
-2
1
0
-3
10
-3
28
25
54 105
88 261
a. p(x) =
b. List all possible rational roots.
c. p(2) =
d. What is the least upper bound?
e. What is the greatest lower bound?
f.
The real roots can be found between what integers?
4. Suppose you saw the following graph on your calculator and knew the information given. Answer the following questions:
Possible rational roots
P N I
1 2 4

4 1 0
 1, 2, 4, , , 
3 3 3

2 1 2
0 1 4
a.
b.
c.
d.
e.
How many roots are there?
How many roots can be found by looking at the graph?
Is the root between 2 and 3 rational or irrational? Explain.
When looking for the other roots, why is it not helpful to find the lower bound?
Is it possible that this polynomial has 4 positive real roots?