Name_____________________________________________________
Date______________________________________________________
PreCalculus - Section 2.4 - Version B
Folder #____
Period______
MC _____/24
CR ______/30
SCORE_____/54
CONSTRUCTED RESPONSE. Show all work and justify your answer in complete sentences by referencing the
appropriate theorem.
1)
A)Use long division to divide f(x) = 3x4 + 12x3 + 43x2 +125x + 11 by x2 + 10.
Write your answer in both fraction form and polynomial equation form.
B) Identify each of the following parts
Divisor = _______________
Dividend = _______________
Quotient = _______________
Remainder = _______________
C) Explain how you would use the remainder theorem to find f(-2).
Use complete sentences!
C-1
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Divide f(x) by d(x), and write a summary statement in the form indicated.
2) f x = x2 - 2x + 3; d x = x - 4 (Write answer in fractional form)
f(x) = x + 2 +
11
A)
B) f(x)
x-4
x-4
x-4
C)
f(x)
x-4
= x-4 +
11
x-4
D)
f(x)
x-4
+
8
x-4
= x+2 +
8
x-4
= x-4
3) f x = x3 + 2; d x = x + 4 (Write answer in polynomial form)
A) f x = x + 4 x2 - 4x + 16 - 62
B) f x = x + 4
C) f x = x + 4
x2 - 4x + 16 - 64
D) f x = x + 4
x2 + 4x + 16 - 62
x2 + 4x + 16 - 64
Divide using synthetic division, and write a summary statement in fraction form.
5
4
2
4) 2x - x + 3x - x + 5
x-1
A)
B)
2x4 + x3 + x2 + 4x + 3 +
8
x-1
2x4 + x3 - x2 + 2x + 1 +
C)
6
x-1
D)
2x4 - 3x3 - x +
6
x-1
2x4 + x3 + 4x2 + 3x +
C-2
8
x-1
4
3
2
5) -5x + x + 2x + 3x - 1
x-1
A) -5x3 + 4x2 - 2x - 1
B) -5x3 + 4x2 + 6x + 9
C) -5x3 - 4x2 - 2x + 1
D) -5x3 + 6x2 - 4x - 1
Find the remainder when f(x) is divided by (x - k)
6) f(x) = x2 + 4x + 5; k = -4
A) -5
B) 5
C) 37
Find the requested function.
7) Find the cubic function with the given table of values.
x
-5 -2 1 7
f x
0 243 0 0
A) f x = x + 5 x - 1 x - 7
C) f x = 3 x + 5 x - 1 x + 7
B) f x = 3 x - 5 x + 1 x + 7
D) f x = 3 x + 5 x - 1 x - 7
C-3
D) 27
CONSTRUCTED RESPONSE. Show all work and justify your answer in complete sentences by referencing the
appropriate theorem.
8)
Given the function f x = x4 - 8x3 + 9x2 + 48x - 90
A)List all of the possible rational zeros of f(x).
B)Use synthetic division to completely factor f(x).
List the following properties of f(x).
The root(s) of f(x) are _________________________________________.
The zero(s) of f(x) are ________________________________________.
The x-intercepts(s) of f(x) are ___________________________________.
The solution(s) of f(x)=0 are ___________________________________.
The factor(s) of f(x) are _______________________________________.
C-4
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the Rational Zeros Theorem to write a list of all potential rational zeros
9) f(x) = 14x3 + 56x2 + 2x - 7
A) ±1, ±1/7, ±2, ±2/7, ±7, ±14
C) ±1, ±1/2, ±7, ±2/7, ±1/14
B) ±1, ±7, ±1/2
D) ±1, ±1/2, ±7, ±7/2, ±1/7, ±1/14
Find all rational zeros.
10) f(x) = 12x3 + 49x2 + 3x - 4
A) - 3, -4, -4
B) - 1 , 1 , -4
3 4
Solve the problem.
11) Find the remainder when x48 + 6 is divided by x + 1.
A) -5
B) 7
C-5
C) - 1 , 1, -4
12
D) - 3, 4, -4
C) 5
D) -7
Find all of the real zeros of the function. Give exact values whenever possible. Identify each zero as rational or
irrational.
12) f x = x3 - 7x2 + 6x + 20
A) 5 (rational), 2 +
B) 5 (rational),
5 (irrational), and 2 -
5 (irrational), and -
5 (irrational)
5 (irrational)
C) 5 (rational), 1 + 5 (irrational), and 1 D) 5 (rational), 6 (rational), -4 (rational)
5 (irrational)
Find the remainder when f(x) is divided by (x - k)
13) f(x) = 4x4 + 6x3 + 6x2 - 5x - 5; k = -2
A) 45
B) -75
C) 102
D) 83
Use synthetic division to determine whether the number k is an upper or lower bound (as specified) for the real zeros
of the function f.
14) k = 1; f x = x4 + 2x3 + 3x2 - 3x + 5; Upper bound?
A) No
B) Yes
C-6
CONSTRUCTED RESPONSE. Show all work and justify your answer in complete sentences by referencing the
appropriate theorem.
15)
Use polynomial theorems to make predictions about the zeros of the given polynomial
p x = 2x5 - 8x4 + 3x +25 .
Use a sentence frame to state the answers in mathematically and
grammatically correct sentences. DO NOT FIND THE ZEROS.
A) Use the Intermediate Value Theorem (IVT) to conclude if possible whether or not the function
has a zero on the interval [-2,0], [0,3], [3,5].
B) Use x = -3 and x = 6 with the bounds tests to give upper and lower bounds for the zeros of p(x).
C-7
C) Use Descartes’ Rule of Signs to make a prediction about the number of positive and negative
real zeros for p x = 2x5 - 8x4 + 3x +25.
C-8
Answer Key
Testname: P.C2.S4.C
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
A
A
A
C
B
D
f(x) = x + 2
D
B
B
C
A
B
x-6
7x - 11
C-9