M3Sem2Rev#1_2018 Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Which is a graph of an even function with a positive leading coefficient?
A C
B D
____ 2. If f(x) is an odd function with a negative leading coefficient, g(x) is an even function with a negative leading coefficient, and h(x) is the product of f(x) and g(x), which of the following could be the graph of h(x)?
A C
____ 3. Which of the following polynomial functions could have the graph shown?
1 2 3 4 5 –1
–2 –3 –4
–5 x
y
A B
C D
____ 4. Which of the following is a true statement about the graph of ?
A It passes through the x-axis once and is never tangent to the x-axis. B It passes through the x-axis once and is tangent to the x-axis once. C It passes through the x-axis twice and is tangent to the x-axis once. D It passes through the x-axis three times and is never tangent to the x-axis.
____ 5. Multiply .
A C
B D
____ 6. Which divisor of results in a remainder of 86? A
B C D
____ 7. Which of the following is a factor of ? A
B C D x +2
____ 8. Which of the following is NOT a factor of ?
____ 9. If is a factor of a polynomial , which of the following statements does NOT have to be true?
A C 2 is a root of .
B D 2 is a zero of
____ 10. Use the remainder theorem to determine the remainder when is divided by .
A B C D 168
____ 11. The quotient when is divided by is . Which of the following is not
necessarily true?
A
B is not a zero of . C is not a factor of .
D Substituting 1 for in results in .
____ 12. Write an equivalent expression for .
A C
B D
____ 13. Write an equivalent expression for .
A C
B D
____ 14. Expand . A
B
C
D
____ 15. Expand . A
B
C
D
____ 16. Jon has rewritten the expression in order to factor it. Describe a reasonable next step for Jon to perform. A Use the Commutative Property to rewrite the terms in a different order.
B Factor 7 from the second and fourth terms.
____ 17. What is the complete factorization of ?
A C
B D
____ 18. Completely factor .
A C
B D cannot be factored
____ 19. Which of the following is equal to ?
A B
C
D
____ 20. Idnetify the roots of . State the multiplicity of each root. A The root 2 has a multiplicity of 1, and the root –3 has a mulitplicity of 1. B The root –2 has a multiplicityof 1, and the root –3 has a mulitplicity of 2. C fThe root 2 has a multiplicityof 1, and the root 3 has a mulitplicity of 1. D The root 2 has a multiplicityof 1, and the root –3 has a mulitplicity of 2.
____ 21. Which of the following lists all the roots of ?
A C
B D
____ 22. Which is a list of all the roots of ?
A , 0, 4 C –4, 7
B , 0, 7 D –4, 1, 7
____ 23. Which is a third degree polynomial with and 1 as its only zeros?
A C
B D
____ 24. If 3 and are two of the roots of a third degree polynomial with integer coefficients, which of the following is the other
root?
A C
____ 25. Which is a list of all the roots of ?
A 0, 2 C , 0, 2
B , 0, 9 D , , 0, 2, 9
____ 26. Which is a third degree polynomial with and 2 as its only zeros?
A C
B D
____ 27. Which is a list of all the roots of ?
A , 2, C , ,
B 1, 2, D 1, ,
____ 28. The graph of is shown below. The graph of can be obtained by applying horizontal and vertical shifts to the parent
function . What is ?
2 4 6
–2 –4
–6 x
2 4 6
–2 –4 –6
y
A
B
C
D
____ 29. Find the average rate of change of the function from to .
A B
C
____ 30. The graph of which function is shown?
2 4 6
–2 –4
–6 x
2 4 6
–2 –4 –6
y
A
B
C
D
Multiple Response
Identify one or more choices that best complete the statement or answer the question.
____ 1. Which of the following statements are true about the polynomial function ? (The zeros of are integers, and the graph of
does not cross the -axis at places other than those shown.)
y = p(x)
2 4 6 8
–2
–4 x
150 300 450
–150 –300 –450
y
A The degree of is even.
B The degree of is 4.
C The leading coefficient of is negative.
D The degree of is at least 6.
E The graph of has a -intercept of 150.
____ 2. Which of the following expressions can be rewritten as a sum of cubes, a difference of cubes, or a difference of squares?
A B
C
D E
F
____ 3. Rational expressions sometimes have excluded values because these values make the expression undefined. The excluded values can sometimes lead to extraneous solutions when solving an equation that contains the expression. Which of the following rational expressions have excluded values?
A
B
C
D
E
____ 4. Choose all the statements that are true about the graph.
2 4 6 8 10 12 –2
–4 x
0.5
–0.5 –1 –1.5 –2 –2.5 –3 –3.5
f(x)
A The x-intercept is 9, B The -intercept is .
C is increasing when .
D is decreasing when .
E has a local maximum at .
F
has a local minimum at .
G is negative when .
Matching
Match each polynomial function to the correct description of its graph.
A Passes through the horizontal axis once and is never tangent to the horizontal axis
D Does not pass through the horizontal axis
B Passes through the horizontal axis twice and is never tangent to the horizontal axis
E Is tangent to the horizontal axis once and never passes through the horizontal axis
C Passes through the horizontal axis three times and is never tangent to the horizontal axis
F Passes through the horizontal axis once and is tangent to the horizontal axis once
____ 1.
____ 2.
Short Answer
1. Expand .
2. Expand .
3. is a polynomial equation.
Part A: How many roots are there for ? Explain how you determined your answer.
Part B: How many of those roots are real roots? Justify your answer. Include a graph if necessary.
4. Simplify.
M3Sem2Rev#1_2018 Answer Section MULTIPLE CHOICE
1. ANS: B PTS: 1 DIF: DOK 1 NAT: F-IF.B.4
STA: F-IF.4
2. ANS: A PTS: 1 DIF: DOK 3 NAT: F-BF.B.3 | F-IF.B.4 STA: F-BF.3 | F-IF.4
3. ANS: D
The function has 3 zeros, , 2, and 5. Since is a factor twice, is a zero twice, so the function’s graph would be tangent to the x-axis at . Since and are factors once, 2 and 5 are zeros once, so the function’s graph would cross the x-axis at and .
Feedback
A Make sure you have identified the zeros of the function correctly.
B Make sure you have identified the zeros of the function correctly.
C This function is cubic, so its end behavior will not be the same at both ends of the graph.
D That’s correct!
PTS: 1 DIF: DOK 1 NAT: A-APR.B.3 STA: A-APR.3 KEY: zeros of polynomial functions | graphs of polynomial functions
4. ANS: B
The number of times the graph of a polynomial function intersects the horizontal axis is equal to the number of unique zeros the polynomial has.
This polynomial has 2 unique zeros, 6 and , so the graph intersects the x-axis twice. Since one of those zeros, 6, is a zero twice, the graph is tangent to the x-axis at .
Feedback
A Use the zeros to sketch a graph of the polynomial function.
B That’s correct!
C Use the zeros to sketch a graph of the polynomial function.
D Use the zeros to sketch a graph of the polynomial function.
PTS: 1 DIF: DOK 1 NAT: A-APR.B.3 STA: A-APR.3 KEY: zeros
5. ANS: B PTS: 1 DIF: DOK 1 NAT: A-APR.A.1
STA: A-APR.1
6. ANS: A PTS: 1 DIF: DOK 1 NAT: A-APR.B.2
STA: A-APR.2
7. ANS: A PTS: 1 DIF: DOK 1 NAT: A-APR.B.2
STA: A-APR.2
8. ANS: D PTS: 1 DIF: DOK 2 NAT: A-APR.B.2
STA: A-APR.2
STA: A-APR.2
10. ANS: B
The remainder theorem says that for a polynomial and a number a, the remainder on division of by is .
Substitute for in and simplify to find the remainder.
Feedback
A It seems you squared a negative number and got a negative result when finding . B That’s correct!
C You found , not . D You found , not .
PTS: 1 DIF: DOK 1 NAT: A-APR.B.2 STA: A-APR.2 KEY: remainder theorem
11. ANS: B
The remainder theorem says that for a polynomial and a number a, the remainder on division of by is . The factor theorem states that if and only if is a factor of . Since did not divide evenly, it is not a
factor of and . Also, since the remainder of the polynomial division is , it is true that substituting 1 for in results in .
While it is true that 1 is not a zero of , it is not necessarily true that is not a zero of ; in fact, in this case is a zero of .
Feedback
A Since did not divide evenly, . B That’s correct!
C Since did not divide evenly, is not a factor of . D The quotient shows that the remainder when is divided by is .
PTS: 1 DIF: DOK 2 NAT: A-APR.B.2 STA: A-APR.2 KEY: remainder theorem | factor theorem
12. ANS: A PTS: 1 DIF: DOK 1 NAT: A-APR.C.4
STA: A-APR.4 KEY: polynomial identities
13. ANS: D PTS: 1 DIF: DOK 1 NAT: A-APR.C.4
STA: A-APR.4 KEY: polynomial identities
14. ANS: D PTS: 1 DIF: DOK 1 NAT: A-APR.C.5
STA: A-APR.5
15. ANS: A PTS: 1 DIF: DOK 1 NAT: A-APR.C.5
STA: A-APR.5 TOP: Use Combinations and the Binomial Theorem KEY: binomial theorem | expand
16. ANS: C PTS: 1 DIF: DOK 2 NAT: A-SSE.A.1b
STA: A-SSE.1b KEY: complicated expressions
17. ANS: C PTS: 1 DIF: DOK 1 NAT: A-SSE.A.2
18. ANS: B PTS: 1 DIF: DOK 1 NAT: A-SSE.A.2 STA: A-SSE.2
19. ANS: B
Notice that and , so .
The given expression is a difference of two squares. The factors of a difference of two squares are the sum of the roots and the
difference of the roots, .
Feedback
A Notice that the given expression is a difference of two squares.
B That’s correct!
C Notice that the given expression is a difference of two squares.
D Notice that the given expression is a difference of two squares.
PTS: 1 DIF: DOK 1 NAT: A-SSE.A.2 | MP.7
STA: A-SSE.2 | MP.7 KEY: rewriting expressions | properties of exponents
20. ANS: D PTS: 1 DIF: DOK 1 NAT: A-APR.B.3
STA: A-APR.3
21. ANS: C PTS: 1 DIF: DOK 2 NAT: A-APR.B.3
STA: A-APR.3
22. ANS: A PTS: 1 DIF: DOK 1 NAT: A-APR.B.3
STA: A-APR.3
23. ANS: A PTS: 1 DIF: DOK 2 NAT: A-APR.B.3
STA: A-APR.3
24. ANS: B PTS: 1 DIF: DOK 2 NAT: A-APR.B.3
STA: A-APR.3
25. ANS: C PTS: 1 DIF: DOK 2 NAT: A-SSE.B.3 | A-APR.B.3 STA: A-SSE.3 | A-APR.3
26. ANS: C PTS: 1 DIF: DOK 2 NAT: A-APR.B.3
STA: A-APR.3
27. ANS: D PTS: 1 DIF: DOK 1 NAT: N-CN.C.8 | A-SSE.B.3 | A-APR.B.3 STA: N-CN.8 | A-SSE.3 | A-APR.3
28. ANS: B
2 4 6 –2
–4
–6 x
2 4 6
–2 –4 –6
y
Compare the graph of to the graph of . The graph of can be shifted left 2 units and down 4 units to obtain the graph
of . So, .
Feedback
A Double check the signs of the numbers representing the horizontal and vertical shifts.
B That’s correct!
C Double check the signs and placement of the numbers representing the horizontal and vertical shifts.
D Double check the placement of the numbers representing the horizontal and vertical shifts.
PTS: 1 DIF: DOK 1 NAT: F-BF.B.3 STA: F-BF.3
KEY: cube root function | horizontal shifts | vertical shifts | transformations
29. ANS: C
B
You computed . Compare this to the formula for the average rate of change of a function.
C That’s correct! D
You computed . Compare this to the formula for the average rate of change of a function.
PTS: 1 DIF: DOK 1 NAT: F-IF.B.6* STA: F-IF.6* KEY: average rate of change from a function rule
30. ANS: C
The given graph is a translation of the graph of the parent cube root function 2 units to the left. So, the function is .
Feedback
A The graph shown is not the graph of a square root function.
B The graph shown is not the graph of a square root function.
C That’s correct!
D The given graph is a horizontal translation, not a vertical translation, of the graph of the parent cube root function.
PTS: 1 DIF: DOK 1 NAT: F-IF.C.7b* | F-BF.B.3
STA: F-IF.7b* | F-BF.3 KEY: cube root functions | horizontal translation
MULTIPLE RESPONSE 1. ANS: A, D, F
A, C: In the graph, notice that approaches as approaches and as approaches . Since the end behavior is
the same as approaches and as approaches , the degree of is even. Since approaches at both ends, the leading coefficient is positive.
B, D, F: In the graph, notice that crosses the -axis at and and is tangent to the -axis at and
. So, has zeros of even multiplicity at and and zeros of odd multiplicity at and
. Since has two zeros of even multiplicity and two zeros of odd multiplicity, its degree is at least 6.
E: In the graph, passes through the point . So, the graph of has a -intercept of .
Feedback Correct That’s correct!
Incorrect Examine the key properties of the graph of .
PTS: 2 DIF: DOK 2 NAT: F-IF.C.7c* STA: F-IF.7c* KEY: graphs of polynomial functions | intercepts | zeros | end behavior
2. ANS: A, B, D, F
A: can be rewritten as a difference of perfect squares, .
B: can be rewritten as a sum of perfect cubes,
C: cannot be rewritten because is not a perfect square or a perfect cube.
D: can be rewritten as a difference of perfect squares, .
E: cannot be rewritten because is not a perfect square or a perfect cube.
Feedback Correct That’s correct!
Incorrect Use the properties of exponents to find how each expression can be rewritten.
PTS: 2 DIF: DOK 2 NAT: A-SSE.A.2 STA: A-SSE.2
KEY: rewriting expressions | properties of exponents | difference of squares | difference of cubes | sum of cubes
3. ANS: A, B, D
A: The denominator has a value of 0 when and . So, 4 and 6 are excluded values.
B: The denominator has a value of 0 when . So, is an excluded value.
C: The denominator is never equal to 0, so there are no excluded values.
D: The denominator has a value of 0 when . So, 2 is an excluded value. E: The denominator is never equal to 0, so there are no excluded values.
Feedback Correct That’s correct!
Incorrect Determine which expressions have restrictions on the variable.
PTS: 2 DIF: DOK 1 NAT: A-REI.A.2 STA: A-REI.2 KEY: extraneous solutions | domain restrictions
4. ANS: A, C, G
(9, 0)
(0, –3) (1, –2)
2 4 6 8 10 12 –2
–4 x
0.5
–0.5 –1 –1.5 –2 –2.5 –3 –3.5
f(x)
A: As labeled on the graph, the x-intercept is . B: The -intercept is .
C: is always increasing, so is increasing when .
D: is always increasing, so is not decreasing when .
E: has no local maxima.
F: has no local minima.
G: is below the -axis when , so is negative when .
H: , so is not positive when .
Feedback Correct That’s correct!
PTS: 2 DIF: DOK 1 NAT: F-IF.B.4* STA: F-IF.4*
KEY: cube root function | graph | increasing | decreasing | intercepts | positive function | negative function
MATCHING
1. ANS: E PTS: 1 DIF: DOK 1 NAT: A-APR.B.3
STA: A-APR.3 KEY: zeros
2. ANS: A PTS: 1 DIF: DOK 1 NAT: A-APR.B.3
STA: A-APR.3 KEY: zeros
SHORT ANSWER 1. ANS:
PTS: 1 DIF: DOK 1 NAT: A-APR.C.5 STA: A-APR.5
2. ANS:
PTS: 1 DIF: DOK 1 NAT: A-APR.C.5 STA: A-APR.5 KEY: binomial theorem
3. ANS:
Part A: Because the polynomial has degree 5, there are 5 roots.
Part B:
has three real roots (one of these is a double root) and two complex roots.
Sample explanation:
There is a common factor, :
So, or
means there is a double root at .
is a cubic function, which either has 3 real zeros or one real zero and two complex zeros.
Graphing (see below) shows that there is one real zero.
Therefore, has three real roots (one of these is a double root) and two complex roots.
1 2 3 4 5
–1 –2 –3 –4
–5 x
1 2 3 4 5
–1
–2
–3
–4
–5
PTS: 1 DIF: DOK 3 NAT: N-CN.C.9 STA: N-CN.9
4. ANS:
PTS: 1 DIF: DOK 1 NAT: A-APR.D.7 STA: A-APR.7 LOC: NCTM.PSSM.00.MTH.9-12.ALG.2.c KEY: reciprocal
5. ANS:
