Algebra 2
TEST REVIEW: Unit 3 - Polynomial Functions and Equations
Name ______________________________
Date ____________________ Block ______
“It is not enough to have a good mind; the main thing is to use it well.” –Rene Descartes
Unit 3 Learning Targets
Set A
I can simplify polynomial expressions and apply the properties of exponents.
Set B
I can divide polynomials using polynomial long division and synthetic division and apply the
properties of the Remainder and Factor Theorems.
Set C
I can describe the characteristics and behavior of a polynomial function given its graph.
Set D
I can write the equation of a polynomial function given its zeros/roots or graph.
Set E
I can solve a higher degree polynomial equation over the set of complex numbers by factoring.
Set F
I can find the zeros of a higher degree polynomial function over the set of complex numbers using
the process of depressing a polynomial.
Learning Target Set A (3 points each)
2𝑎 3 𝑏−6
1. Simplify (-4x ) (7x ).
2. Simplify 6𝑎2 𝑏−8 . Assume no variable equals zero.
3. Simplify -6v3w4 (2v3w2 + 3v5w3).
4. If f(x) = x2 – 3x + 7, find f(3b3).
5 2
0
Learning Target Set B (4 points each)
5-6: Circle the best solution.
5. Given that (x - 2) is a factor of
6. Find the quotient using long division. Your work must be shown to
x4 + 5x2 – 36, which of the
receive full credit.
(x3 + 2x2 – 5x – 6 ) ÷ (x + 3)
following is the remainder when
x4 + 5x2 – 36 is divided by (x - 2)?
A.
B.
C.
D.
-2
0
2
-18
Learning Target Set C
7-10: Circle the best solution. (3 points each)
7. Which is the maximum number 8. Which is the degree and sign of the
of turns for a function of the
leading coefficient of the polynomial
form: f(x) = ax5 + bx2 + cx + d
function shown at right?
A.
B.
C.
D.
4
3
2
1
A.
B.
C.
D.
degree 4, positive
degree 3, positive
degree 4, negative
degree 3, negative
9. Approximate the intervals for which the function
is increasing.
10. Which best describes the end behavior of the
function shown?
y
20
10
(1,9)
(4,
x
(2,0)
(-2,0)
-4
(0,0)
-2
2
4
-10
-20
-30
A.
B.
C.
D.
A. As x  -  , f(x)
B. As x  -  , f(x)
C. As x  -  , f(x)
D. As x  -  , f(x)
(-  ,  )
(-1, 1)
(-  , -1), (1,  )
(-  , 0), (0,  )
 -  and as x   , f(x)  - 
  and as x   , f(x)  
 -  and as x   , f(x)  
  and as x   , f(x)  - 
11. Use the graph at the right to complete a - h. (8 points)
a. Is the function of even or odd degree? How do you know?
y
40
(0,20)
b. Is the leading coefficient positive or negative? How do you know?
20
(-5,0)
c. Estimate the real zeros of the function.
-6
(-2,0)
-4
(2,0)
-2
d. Does there appear to be any multiple zeros?
How do you know?
2
-20
-40
f. Is the function increasing or decreasing on the interval x: (2,  ).
g. What is the smallest possible degree of the
function? How do you know?
h. State the domain and range of the function.
Domain:
Range:
12. Complete a – h for f (x) = x(x +3)2(x - 1)(x2 + 9). (10 points)
What is the degree of this function?
h. Sketch the graph of this function without using a
a.
HINT: the function is in factored form.
calculator. Be sure to label the units on the x-axis.
What is the maximum number of
b.
turns in this function?
How many total zeros does this
c.
function have?
What are the real zeros of this
d.
function?
Are there any points of tangency to
e.
the x-axis? If so, at what x-value?
Are there any terrace points at the x-f.
axis? If so, at what x-value?
g. As x  -  , f(x)  ______ and as x   , f(x)  ______
x
4
Learning Target Set D (4 points each)
13-14: Circle the best solution.
13. Which defines a polynomial
14. Which of the following could
function with zeros
be the function for the
4, -1, and -2?
graph at the right?
A. f (x) = x(x – 4)(x + 1)(x + 2)
A. p(x) = -x(x – 3)2
B. f (x) = x(x + 4)(x – 1)(x – 2)
B. p(x) = x(x + 3)
C. f (x) = (x – 4)(x + 1)(x + 2)
C. p(x) = -x(x – 3)3
D. f (x) = (x + 4)(x – 1)(x – 2)
D. p(x) = x(x + 3)3
15: Write the equation of the fifthdegree polynomial function in
its factored form that has
a = -3, a tangent point at 3,
and other roots of ±2i and 7.
16. Write the polynomial function in factored form for the graph.
(0, -9)
15. __________________________
Learning Target Set E
17-18: (3 points each)
17. Factor completely: 64𝑥 3 − 8.
16. ________________________________________
18. Solve by factoring: 2x3 = -16. Write the following is the factored
form and express the solution set?
19-21: SOLVE each polynomial over the set of complex numbers by FACTORING. (GCF, factor by grouping…etc)
(Provide exact solutions only, no decimal answers.) (5 points each)
20. x3 + 4x2 – 3x = 12
21. 2x4 – 200x2 = 0
19. 𝒙𝟐 − 𝟐𝒙 − 𝟖 = 𝟎
factored form:
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factored form:
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factored form:
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solutions:
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solutions:
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solutions:
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Learning Target Set F (6 points each)
22-23: Circle the best solution.
22. Which describes the number and type of roots of
the equation m4 + 100 = 0 ?
23. Which is the solution set for the equation
x4 – 11x2 = 80 ?
A.
B.
C.
D.
A.
B.
C.
D.
4 real roots
3 real roots with one double root
4 imaginary roots
2 real roots and 2 imaginary roots
24. Given the function f(x) = 2x3 + 5x2 – 4x – 3
find all of the roots using synthetic division.
Hint: Divide out the whole number zeros from the graph
(Provide exact solutions only, no decimal solutions.)
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{4i , -4i , i 5 , -i 5 }
{4 , -4 , i 5 , -i 5 }
{4i , -4i , 5i, -5i}
{4 , -4 , 5, -5}
BONUS:
Given the function f(x) = x3 - 5x2 - 7x + 35, find all
of the roots of the function.
(Provide exact solutions only, no decimal solutions.)
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