Name________________________________
Polynomial Review
Simplify.
1.
(40 w2 ) 5
2. (3 y 4 z 2 )( y 3 z 5 )
3.
12a 3b9
21a 2b 5
4.
w3 (v 5 )2

v 5 ( w3 )2
5. (10v 4  2v 2  6v3  7)  (9  v  2v 4 )
6. (4t  1)2
7. ( z  5)3
8. (5q  2)(8q  1)(q  4)
8. (2 f  1)3
10. ( x3  2 x  1)( x3  x 2  5)
11. What is the value of the function f ( x)  2 x5  3x 4  x3  6 x  20 when x  2 .
Factor.
12. x 3  125
13. 64n3  27
14. 2 w3  54
15. 24q3  81
16. The dimensions (in inches) of a jewelry box are: length 2x , width ( x  1) , and height ( x  3) . If the volume of the
box is 24 cubic inches, find the dimensions of the box.
Divide.
16. ( x3  x  30)  ( x  3)
17. (5 x 4  2 x3  9 x  12)  ( x 2  3x  4)
Write a polynomial function f of least degree with rational coefficients, a leading coefficient of 1 and the given zeros.
18.
1, 2, 6
19. 8, (2  i )
20. 5, 3
21. 2i, 5i
22. List the possible rational zeros of the function g ( x)  9 x5  3x3  7 x  4
Analyze the polynomial without a calculator. Then make a possible sketch of the graph.
23. g ( x)  x3  8 x 2  15 x  54
24. h( x)  x3  6 x 2  x  6
Degree__________________
Odd/Even________________
Pos/Neg_________________
Possible turns_____________
Real Roots-Describe the nature
_________________________
_________________________
Imaginary Roots____________
Actual Turning points________
Degree__________________
Odd/Even________________
Pos/Neg_________________
Possible turns_____________
Real Roots-Describe the nature
_________________________
_________________________
Imaginary Roots____________
Actual Turning points________
f ( x)  ______ as x  
f ( x)  ______ as x  
f ( x)  ______ as x  
f ( x)  ______ as x  
25. h( x)  x 4  x 2  20
26. h( x)  2 x 4  x3  x 2  x  1
Degree__________________
Odd/Even________________
Pos/Neg_________________
Possible turns_____________
Real Roots-Describe the nature
_________________________
_________________________
Imaginary Roots____________
Actual Turning points________
Degree__________________
Odd/Even________________
Pos/Neg_________________
Possible turns_____________
Real Roots-Describe the nature
_________________________
_________________________
Imaginary Roots____________
Actual Turning points________
f ( x)  ______ as x  
f ( x)  ______ as x  
f ( x)  ______ as x  
f ( x)  ______ as x  
Determine the lowest-degree polynomial that has the given graph.
27.
28.
29.
Sketch a graph of the polynomial function.
30. g ( x)  ( x  1)( x  1)3 ( x  2) 2
f ( x)   x 2 ( x  3)( x  2)3
31.
y
y
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
x
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x
32. Write a cubic function that passes through the points (1, 0), (2, 0), (3, 0), (0, 6)
33. Write the polynomial equation that fits the data:
x
1
2
3
4
5
6
f(x)
-1
3
3
5
15
39
34. Sketch a graph of a polynomial with the given zeros
and corresponding multiplicities.
y
(Note: Graph is not unique)
8
x = -7, multiplicity of 1
x = -1, multiplicity of 2
x = 3, multiplicity of 3
x = 6, multiplicity of 1
46. Give a possible factorization of the
polynomial. Do Not multiply out the
factors.
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
–6
–7
–8
1 2 3 4 5 6 7 8
x
(9,9)
(-2,0)
(0,-6)
(4,0)
(10,0)
(11,-7)
+
1. x-intercept(s):____________ ____
2. y-intercept:________________
3. Function?__________________
4. Domain:_____________________
5. Range:____________________
6. Find f(11) ___________________
7. Where does f(x) equal zero? _________________
8. Where is f(x) < 0?_______________________________
9. Where is f(x) > 0?__________________________
10. How many times does the line y = -2 intersect?_______
11. Where does f(x) = 9?_________
12. Absolute maximum:_________
13. Absolute minimum: _________
14. Relative Maximum:__________
15. Relative minimum:__________
16. Even or odd function:________
17. (+/-) leading coefficient: ______
18.
19. Where is the graph increasing?_______________ 20. Where is the graph decreasing?____________________
21. Number of turning points:______________
22. What do you think the degree of the function is?______