Name________________________
Date ________
Factoring Polynomials by GCF, Sum/Difference of Cubes,
Operations with Polynomials, Composition of Functions
25
Complete the 5 review questions and then choose 20 POINTS worth of new material.
Mix and match as YOU NEED to feel confident with this material.
I.
REVIEW – Factor all of these completely (1 pt each)
1.
16x  y
3.
8 x 2  50 y 2
2
2
2.
4.
______
25 x  1
2
25
3x3  10 x 2  8 x
5.
2 x2  8x  8
II. PRACTICE – Factor or Simplify (1 pt each)
1. (7q  3q3) + (16  8q3 + 5q2  q)
2. (4z4 + 6z  9) + (11  z3 + 3z2 + z4)
3. (l0v4  2v2 + 6v3  7)  (9  v + 2v4)
4. (4x5 + 3x4  5x + l)  (x3 + 2x4  x5 + 1)
5. 2x3(5x  1)
6. (n + 5)(2n2  n  7)
7. Factor x3 – 27
8. Factor x3 + 27
9. Factor x3 – 64
10. Factor 2x3 + 16
III. MORE PRACTICE – 2 pts each
11.
8 x3  27
14.
8m  64n
3
12.
8 x3  27
13.
15. Multiply (x – 3)3
64 x 3  1
16. Multiply (2x + 5)3
3
17. f(x) = 3x – 5, g(x) = 2x + 1, Find f o g
18. f(x) = x – 2, g(x) = x2 + 3, Find g o f
19. Given f(x) = 2x2, g(x) = x2 + 7. a) find f(g(-1))
20. f(x) = x2 + 3x and g(x) = x2 + 6. a) find f(g(-2))
b) g(f(-2))
b) find g(f(-2))
IV. REACH FOR THE STARS! – 3 pts each
Factor completely.
19.
x6  y 6
20.
(a  b)3  c3
Name________________________
Date ________
25
Algebra2 HW: Evaluating Polynomials with Synthetic Substitution
and Graphing their End Behavior
Complete the 5 review questions and then choose 20 POINTS worth of new material.
Mix and match as YOU NEED to feel confident with this material.
II.
REVIEW – Simplify (1 pt each)
1.
1
4
2 x  3x
1
3
2.
3. Factor and solve 3x2 - 27
6
4
25
3 1
2 6
(x y z )
4. Factor and solve 4x2 - 9
5. Factor and solve 2x2 – 2x - 12
II. PRACTICE – Evaluate the polynomial for the given value of x in TWO ways – by direct
substitution and by synthetic substitution (3 pt each)
6. f(x) = 5x3 – 2x2 – 8x + 16
Direct Substitution
for x = 3
7. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9
Direct Substitution
8. f(x) = x3 + 8x2 – 7x + 35
Direct Substitution
9. f(x) = –8x3 + 14x – 35
Direct Substitution
for x = -2
for x = –6
for x = 4
______
Synthetic Substitution
Synthetic Substitution
Synthetic Substitution
Synthetic Substitution
10. f(x) = –2x4 + 3x3 – 8x + 13
Direct Substitution
11. f(x) = –7x3 + 11x2 + 4x
Direct Substitution
for x = 2
Synthetic Substitution
for x = 3
Synthetic Substitution
III. MORE PRACTICE – 3 pts each
List the degree of each polynomial and specify if it opens "up" or "down."
Describe the end behavior of the graph of the polynomial function.
Draw a quick sketch on the right of your "model" function.
f(x) = 5x 3
11.
Degree:
Opens:
As x    f(x) ______
12.
and as
x  +  f(x)  ______
f(x) = 2x5  7x2  4x
Degree:
Opens:
As x    f(x)  ______
and as
x  +  f(x)  ______
13. f(x) = 2x8 + 9x7 + 10
Degree:
Opens:
As x    f(x)  ______
and as
x  +  f(x)  ______
14. f(x) =  12x6  2x + 5
Degree:
As x    f(x) ______
Opens:
x  +  f(x)  ______
and as
IV. REACH FOR THE STARS – 4 pts each
15. f(x) = 2013x80 - 95x50 + 407x
Degree:
Opens:
16. f(x) = 2017x57  1998x46 + 1999x23
Degree:
Opens:
As x    f(x)  ______
As x    f(x)  ______
x  +  f(x)  ______
x  +  f(x)  ______
Name________________________
Date ________
25
Algebra2 HW: Fundamental Theorem of Algebra and Graphing Polynomials
Complete the 5 review questions and then choose 20 POINTS worth of new material.
Mix and match as YOU NEED to feel confident with this material.
I.
1.
4.
______
REVIEW – Factor (remember the GCF!) (1 pt each)
x3  125
2.
y4  8y
27g3  343
25
3.
5.
64n3  27
40v3  625
II. PRACTICE – Identify the number of solutions and maximum possible number of turning
points (not that the function HAS that many, but the most possible). (1 pt each)
6. f(x) = 5x3 – 2x2 – 8x + 16
7. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9
ZEROS:_________
ZEROS:_________
Max # of turning points:________
Max # of turning points:________
8. f(x) = x7 + 8x4 – 7x + 35
ZEROS:_________
Max # of turning points:________
9. f(x) = –8x3 + 14x – 35
ZEROS:_________
Max # of turning points:________
10. f(x) = –2x6 + 3x3 – 8x + 13
ZEROS:_________
Max # of turning points:________
III. MORE PRACTICE – Write a polynomial function in standard form of least degree that has a
leading coefficient of 1 and the given zeros. (Remember, imaginary and irrational solutions
always come in pairs!) (2 pts each)
11. -2, 1, 3
12. -5, -1, 2
13. 2, -i, i
15. 4,  5 ,
14. 2, -3i
5
16. 3, 1  2
EVEN MORE PRACTICE – Answer the questions about the polynomials (3 pts/problem)
17. a. The Number of real zeros for the polynomial _________
Name them __________
b. The Number of turning points __________
c. Intervals on which the function is increasing
_______________, ________________
d. Intervals on which the function is decreasing
_______________, ________________
e.
Does the graph have a relative minimum(s)? ______
If so, name the point. _______
f.
Does the graph have a relative maximum(s)? ______
If so, name the point. _______
g.
Does the graph have an absolute minimum? _________
If so, name the point. _______
h. Does the graph have an absolute maximum? _________
If so, name the point. _______
18.
i.
The Number of real zeros for the polynomial _________
Name them __________
j.
k.
The Number of turning points __________
Intervals on which the function is increasing
_______________, ________________
l.
Intervals on which the function is decreasing
_______________, ________________
m. Does the graph have a relative minimum(s)? ______
If so, name the point. _______
n.
Does the graph have a relative maximum(s)? ______
If so, name the point. _______
o.
Does the graph have an absolute minimum? _________
If so, name the point. _______
p.
Does the graph have an absolute maximum? _________
If so, name the point. _______
III. REACH FOR THE STARS (4 pts)
19. Graph a polynomial of degree 5 with a negative leading coefficient, zeros at 2, 4, -1, and turning points at
x=0 and x=3. How many local max's will your graph have? ________
How many local mins?________ Will your graph have a global maximum? Why or why not?
 y
Name the intervals on which your function is increasing:



The intervals on which it is decreasing:

    
Describe the function's end behavior:



Explain how the function can have degree 5 but only 3 real zeros.


x





Quiz Review: Analyzing Graphs
Answer the questions based on the graph to the left.
Degree: (even or odd) _____________
Leading Coefficient: (positive or negative) ___________________
End Behavior: As x  , f ( x)  ____
9 y
8
7
6
5
4
3
2
1
-1
-9-8 -7 -6 -5-4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
-9
As x  , f ( x)  ____
Identify the real zeros of the graph:
___________________________________________
x
Identify the factors based on your zeros listed above.
1 2 3 4 5 6 7 8 9
___________________________________
Circle the turning points on the graph. Determine if they are relative
maximums or minimums, absolute maximums or minimums.
Determine the intervals where the polynomials are
Increasing: ______________________________
Decreasing: _____________________________
Determine the domain and range of the polynomial.
Domain: _____________
Range: _______________
Degree: (even or odd) _____________
Leading Coefficient: (positive or negative) ___________________
End Behavior: As x  , f ( x)  ____
9 y
8
7
6
5
4
3
2
1
-1
-9-8 -7 -6 -5-4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
-9
As x  , f ( x)  ____
Identify the real zeros of the graph:
___________________________________________
x
1 2 3 4 5 6 7 8 9
Identify the factors based on your zeros listed above.
___________________________________
Circle the turning points on the graph. Determine if they are relative
maximums or minimums, absolute maximums or minimums.
Determine the intervals where the polynomials are
Increasing: ______________________________
Decreasing: _____________________________
Determine the domain and range of the polynomial.
Domain: _____________
Operations & Substitution
Range: _______________
Simplify each expression. Show work!
1. (2x + 5)2
2. (2x4 − 8x2 − x) − (−5x4 − x + 5)
3. (5d3 – 4d2 + 5) + (7d3 + 2d2 – 8d)
4. (x + 5i)(x – 5i)
5. (2x + 3)(x – 2)(3x + 2)
6. (x + 4)(x2 + 2x – 3)
7. Evaluate the polynomial function using Direct Substitution.
f(x) = -3x3 + x2 – 12x – 5 when x = -2
8. Evaluate the polynomial function using Synthetic Substitution.
F(X) = x4 + 2x3 + 5x - 8 for f(-4)
9. Write a polynomial function that has real coefficients, the given zeros, and a leading coefficient of 1.
Zeros: 2, 4, -3i
Recognizing and "Reading" Polynomials
Identify the degree, leading coefficient, and constant of the polynomial. (State the numerical
value.)
10. f(x) = 6x5 – 4x3 + 1
Degree _____ Leading Coefficient _____
Constant ____
11. g(x) = 9x4 + x – 7
Degree _____ Leading Coefficient _____
Constant ____
12. f(x) = -2x2 – 3x4 + 5x – 9x3 + 5
Degree _____ Leading Coefficient _____
Constant ____
Tell if each of the following are polynomials? If no, explain why not!
(Yes or No)
Explanation
13.
4x2 + 2x + x2 + 3
_________
_____________________
14.
3x + 1
_________
_____________________
15.
5x + 2x½ +3
_________
_____________________
Describe the end behavior of the graph. Use   or  
16. f(x) = -7x2 + 4x – 9
as x   , f ( x)  _____
as x   , f ( x)  _____
17. f(x) = 6x5 + 7x4 – 8x
as x   , f ( x)  _____
as x   , f ( x)  _____
18. f(x) = -8x3 – 9x2 + x – 4
19.
f(x) = x4 + 16
as x   , f ( x)  _____
as x   , f ( x)  _____
as x   , f ( x)  _____
as x   , f ( x)  _____
Factoring
Factor the following completely. If not possible, write PRIME.
20)
2x3 + 16
21)
22)
125x3 - 27
23)
24)
25x2 – 81
25)
x3 – 216
x2 + 4
2x4 + 128x