MTH-112 Quiz 6
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1. Is f (x) = x6 + 16 x +
(yes / no)
1
6
a polynomial function?
4. Is the graph of the above function symmetric
about y axis, origin, or neither? (y axis / origin / neither)
.
If the function is a polynomial, find the following:
.
5. If the graph of a quadratic function has only
one x-intercept, and if the x-intercept is (2, 0),
what is the equation of the axis of symmetry?
(a) Degree:
(b) Leading term:
.
(c) Leading coefficient:
6. Is the following statement true or false?
.
The graph of an odd function always goes
through the origin (0, 0). (Assume that 0 is
in the domain of the function.) (true / false)
2. Find the zeros of f (x) = 11x(x−22)33 (x+44)55 ,
and the multiplicity of each zero.
Zero
Multiplicity
.
7. Can the graph below be a graph of a polynomial
function? (yes / no)
y
4
3
.
f (x)
2
1
3. Find the zeros of f (x) = 5x2 + x3 , and the multiplicity of each zero.
.
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
Zero
Multiplicity
-2
-3
-4
8. Determine the end behavior of the graph of the
function f (x) = 4x7 − 2x6 + 9x3 − 2.
(both up / both down / left up - right down /
left down - right up)
.
.
1
MTH-112 Quiz 6 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. Is f (x) = x6 + 16 x +
( yes / no)
1
6
a polynomial function?
3. Find the zeros of f (x) = 5x2 + x3 , and the multiplicity of each zero.
Since all the exponents of the variable are nonnegative integers, the function is a polynomial.
The zeros are the x values for which f (x) is zero.
Write 0 for f (x), and solve for x.
5x2 + x3 = 0
If the function is a polynomial, find the following:
x2 (5 + x) = 0
5x2 = 0
(a) Degree: 6
5+x=0
x = −5
x=0
The degree is the highest power of the variable.
The factor 5x2 gives the zero 0, and the multiplicity of that zero is the exponent of the factor,
which is 2.
(b) Leading term: x6
The leading term is the term with the highest power of the variable.
The factor 5 + x gives the zero −5, and the multiplicity of that zero is the exponent of the factor,
which is 1.
(c) Leading coefficient: 1
The leading coefficient is the coefficient of
the leading term.
2. Find the zeros of the polynomial function
f (x) = 11x(x − 22)33 (x + 44)55 , and the multiplicity of each zero.
Zero
Multiplicity
0
2
−5
1
4. Is the graph of the above function symmetric
about y axis, origin, or neither? (y axis / origin / neither )
The factor 11x gives the zero 0, and the multiplicity of that zero is the exponent of the factor,
which is 1.
The factor x − 22 gives the zero 22, and the multiplicity of that zero is the exponent of the factor,
which is 33.
The function is neither even nor odd. Therefore,
it is neither symmetric about y− axis nor about
the origin.
The factor x + 44 gives the zero −44, and the
multiplicity of that zero is the exponent of the
factor, which is 55.
5. If the graph of a quadratic function has only
one x-intercept, and if the x-intercept is (2, 0),
what is the equation of the axis of symmetry?
Zero
Multiplicity
0
1
22
33
−44
55
If the graph of a quadratic function has only
one x-intercept, it touches the x−axis and turns
at that x−intercept. Then the vertex is the
x−intercept. So the axis of symmetry is the vertical line through the point (2, 0).
x=2
1
MTH-112 Quiz 6 - Solutions
6. Is the following statement true or false?
8. Determine the end behavior of the graph of the
function f (x) = 4x7 − 2x6 + 9x3 − 2.
The graph of an odd function always goes
through the origin (0, 0). (Assume that 0 is
in the domain of the function.) ( true / false)
(both up / both down / left up - right down /
left down - right up )
7. Can the graph below be a graph of a polynomial
function? (yes / no )
Graphs of polynomial functions are continuous
curves with smooth rounded corners.
y
4
3
f (x)
2
1
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-3
-4
2
MTH-112 Quiz 7
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1. Divide 35x3 − 30x2 − 68x − 12 by 7x + 1 using
long division.
3. Given f (x) = x3 − 4x2 + 9x − 1, find f (−4)
using direct substitution.
.
f (−4) = . . . . . ..
.
4. Given f (x) = x3 − 4x2 + 9x − 1, find f (−4)
using synthetic substitution.
.
The quotient: . . . . . . . . . . . . . . . . . . . . . . . ..
The remainder: . . . . . ..
.
.
2. Divide 11x3 +21x2 −31 by x−3 using synthetic
division.
f (−4) = . . . . . ..
.
5. Determine whether 3 is a zero of
f (x) = x3 − 4x2 + 5x − 3 using synthetic substitution. (yes / no)
.
The quotient: . . . . . . . . . . . . . . . . . . . . . . . ..
The remainder: . . . . . ..
.
1
MTH-112 Quiz 7 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. Divide 35x3 − 30x2 − 68x − 12 by 7x + 1 using
long division.
using direct substitution.
f (−4) = (−4)3 − 4(−4)2 + 9(−4) − 1
5x2 − 5x − 9
7x + 1
= −64 − 64 − 36 − 1
35x3 − 30x2 − 68x − 12
− 35x3 − 5x2
= −165
− 35x2 − 68x
35x2 + 5x
4. Given f (x) = x3 − 4x2 + 9x − 1, find f (−4)
using synthetic substitution.
− 63x − 12
63x + 9
−3
1
−4
2
The quotient: 5x − 5x − 9.
1
−4
9
−1
−4
32
− 164
−8
41
− 165
The remainder: −3.
f (−4) is the remainder, which is −165.
3
2
2. Divide 11x +21x −31 by x−3 using synthetic
division.
11
3
11
21
0
− 31
33
162
486
54
162
455
f (−4) = −165.
5. Determine whether 3 is a zero of
f (x) = x3 − 4x2 + 5x − 3 using synthetic substitution. (yes / no )
1
2
The quotient: 11x + 54x + 162.
3
1
The remainder: 455.
−4
5
−3
3
−3
6
−1
2
3
Since the remainder is not zero, 3 is not a zero of
the function.
3. Given f (x) = x3 − 4x2 + 9x − 1, find f (−4)
1
MTH-112 Quiz 8
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1. Use the function f (x) = 2x3 + x2 − 12x + 9 to
find the following.
2. Let 1 and −3i be two zeros of a third degree
polynomial function f (x), and f (2) = 26.
(a) Find all possible rational zeros of the function.
(a) What is the other zero of the function?
Factor of the leading coefficient.
Factor of the constant term.
(b) Write the factored form of the function
using ‘a’ as the leading coefficient. Then
multiply the factors (FOIL).
f (x) =
(b) Find one actual zero of the function using
synthetic substitution.
(c) Find the value of a, using the given condition f (2) = 26.
(c) Write a factor of the function using the
zero found in part (b).
(d) Factor the function completely using the
factor found in part (c), and the quotient
found in part (b).
(d) Find the polynomial function (do not leave
your answer in factored form, multiply all
factors).
(e) Find all actual zeros of the function using
the factors found in part (d).
1
MTH-112 Quiz 8 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. Use the function f (x) = 2x3 + x2 − 12x + 9 to
find the following.
(a) What is the other zero of the function?
Whenever a complex number is a zero, the
conjugate of that is also a zero.
(a) Find all possible rational zeros of the function.
Other zero: 3i
Factor of the leading coefficient.
Factor of the constant term.
1
3
9
1
±1
±3
±9
2
1
±
2
3
±
2
9
±
2
2
9
(b) Write the factored form of the function
using ‘a’ as the leading coefficient. Then
multiply the factors (FOIL).
Since 1 is a zero, x − 1 is a factor.
Since −3i is a zero, x + 3i is a factor.
Since 3i is a zero, x − 3i is a factor.
Therefore, the factored form is
f (x) = a(x − 1)(x + 3i)(x − 3i)
(b) Find one actual zero of the function using
synthetic substitution.
1
− 12
9
2
3
−9
2
3
x = 1 is a zero.
−9
0
2
1
First, multiply the factors, starting with
(x + 3i)(x − 3i).
f (x) = a(x − 1)(x + 3i)(x − 3i)
= a(x − 1)(x2 − 3ix + 3ix − 9i2 )
= a(x − 1)(x2 − 9(−1))
= a(x − 1)(x2 + 9)
(c) Write a factor of the function using the
zero found in part (b).
= a(x3 − x2 + 9x − 9)
Since x = 1 is a zero, x − 1 is a factor.
(c) Find the value of a, using the given condition f (2) = 26.
x−1
(d) Factor the function completely using the
factor found in part (c), and the quotient
found in part (b).
Replace x with 2, and set it equal to 26.
f (2) = a (2)3 − (2)2 + 9(2) − 9
26 = a(8 − 4 + 18 − 9)
The quotient in part (b) is 2x2 + 3x − 9.
26 = a(13)
f (x) = 2x3 + x2 − 12x + 9
a=2
= (x − 1)(2x2 + 3x − 9)
= (x − 1)(2x − 3)(x + 3)
(d) Find the polynomial function (do not leave
your answer in factored form, multiply all
factors).
(e) Find all actual zeros of the function using
the factors found in part (d).
Replace a with −1 and rewrite the function
found in part (b).
To find zeros, set each factor equal to zero,
and solve for x.
3
x = 1, x = , x = −3
2
f (x) = a(x3 − x2 + 9x − 9)
= 2(x3 − x2 + 9x − 9)
2. Let 1 and −3i be two zeros of a third degree
polynomial function f (x), and f (2) = 26.
= 2x3 − 2x2 + 18x − 18
1
MTH-112 Quiz 9
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
2x2 − 72
to find the following. Write none if necessary.
− x − 12
(a) The equation(s) of the vertical asymptote(s):
(d) The y− intercept (as an ordered pair):
1. Use the rational function f (x) =
x2
(e) The x− intercept(s) (as an ordered pair or ordered pairs):
(b) The equation of the horizontal asymptote:
(c) The equation of the oblique asymptote:
(f) Graph the function. Draw all asymptotes using
dotted lines.
y
11
10
9
8
7
6
5
4
3
2
1
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
1
2
3
4
5
6
7
8
9 10 11
x
MTH-112 Quiz 9 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1. Use the rational function f (x) =
2x2 − 72
to find the following. Write none if necessary.
x2 − x − 12
(a) The equation(s) of the vertical asymptote(s):
the y−intercept is zero, find f (0), which is the
same as the y− value when x is 0.
Set the bottom equal to zero, and solve for x.
x2 − x − 12 = 0
f (0) =
(x − 4)(x + 3) = 0
x−4=0
x+3=0
x=4
y− intercept: (0, 6)
x = −3
(e) The x− intercept (write your answer as an ordered pair):
Vertical asymptotes: x = 4 and x = −3
(b) The equation of the horizontal asymptote:
The x−intercepts are the points where the graph
crosses the x−axis. Since the y−coordinate of
the x−intercepts is zero, write 0 for y, and solve
for x.
The top degree is 2, and the bottom degree is
2. The top degree is equal to the bottom degree.
Therefore the horizontal asymptote is
y=
2(0)2 − 72
−72
=
=6
2
(0) − 0 − 12
−12
top leading coefficient
2
=
bottom leading coefficient
1
2x2 − 72
− x − 12
0 = 2x2 − 72
0=
Horizontal asymptote: y = 2
x2
72 = 2x2
(c) The equation of the oblique asymptote:
36 = x2
None.
±6 = x
(d) The y− intercept (write your answer as an ordered pair):
x−intercepts: (6, 0) and (−6, 0)
The y−intercept is the point where the graph
crosses the y−axis. Since the x−coordinate of
(f) Graph the function. Draw all asymptotes using
dotted lines.
1
MTH-112 Quiz 9 - Solutions
y
10
8
6
4
y=2
2
-10
-8
-6
-4
-2
2
4
-2
-4
-6
-8
-10
x = −3
x=4
2
6
8
10
x