1 Assignment #2 MAC 1140 – Spring 2020 – Due: April 7, 2020 1. We are given the polynomial 𝑓(𝑥) = 𝑥7 − 𝑥6 − 11𝑥5 + 11𝑥4 + 19𝑥3 − 19𝑥2 − 9𝑥 + 9. (a) What is the degree of this polynomial? How many zeros does it have? Do these zeros have to be real and/or distinct? (b) What is the behavior of the polynomial as 𝑥 → +∞ and 𝑥 → −∞? Explain your answer. (c) Describe, in your own words, the rational zero theorem. Taking into account the form of the polynomial, can we use the rational zero theorem to find if the polynomial has rational zeros? If yes, what are the possible zeros? (d) Describe in your own words Descartes’ rule of signs. Using this rule, what is the possible number of positive zeros of the polynomial, and what is the possible number of negative zeros? (e) Calculate the following values: 𝑓(−3) = ________ 𝑓(−2) = ________ 𝑓(−1) = ________ 𝑓(0) = ________ 𝑓(1) = ________ 𝑓(2) = ________ 𝑓(3) = ________ 𝑓(−4) = ________ 𝑓(4) = ________ 2 (f) Describe in your own words the Factor Theorem. Using this theorem show that 𝑓(𝑥) = (𝑥 + 3)(𝑥 + 1)(𝑥 − 1)(𝑥 − 3) ⋅ 𝑔(𝑥). (g) Using long division, find 𝑔(𝑥). Show all your work. (h) Calculate the following values: 𝑔(−1) = ________ 𝑔(1) = ________ (i) Using the Factor Theorem and synthetic division, factor 𝑔(𝑥). Show all your work. (j) Describe the multiplicity of all the zeros of 𝑓(𝑥), and describe the behavior of the graph of 𝑓(𝑥) at these zeros (i.e., is the graph crossing the 𝑥-axis at these zeros or touches and turns around?). 3 (k) What is the maximum number of turning points of 𝑓(𝑥)? (l) Graph 𝑓(𝑥). 4 2. We are given the polynomial 𝑓(𝑥) = 4𝑥5 + 4𝑥4 + 𝑥3 − 2𝑥2 − 2𝑥 + 1. (a) What is the degree of this polynomial? How many zeros does it have? Do these zeros have to be real and/or distinct? (b) What is the behavior of the polynomial as 𝑥 → +∞ and 𝑥 → −∞? Explain your answer. (c) Describe, in your own words, the rational zero theorem. Taking into account the form of the polynomial, can we use the rational zero theorem to find if the polynomial has rational zeros? If yes, what are the possible zeros? (d) Describe in your own words Descartes’ rule of signs. Using this rule, what is the possible number of positive zeros of the polynomial, and what is the possible number of negative zeros? (e) Calculate the following values: 𝑓(−2) = ________ 𝑓(−1) = ________ 𝑓(0) = ________ 𝑓(1) = ________ 𝑓(2) = ________ (f) Describe in your own words the Intermediate Value Theorem. Based on the results in (e) above, and the possible rational zeros described in (c), show that 𝑥 − 1 2 is a factor of 𝑓(𝑥). Clearly describe your reasoning. 5 (g) Describe in your own words the Factor Theorem. Using this theorem show that 𝑓(𝑥) = (𝑥 + 1)(2𝑥 − 1) ⋅ 𝑔(𝑥

1. 1
Assignment #2
MAC 1140 – Spring 2020 – Due: April 7, 2020
1. We are given the polynomial �(�) = �7 − �6 − 11�5 +
11�4 + 19�3 − 19�2 − 9� + 9.
(a) What is the degree of this polynomial? How many zeros
does it have? Do these zeros have
to be real and/or distinct?
(b) What is the behavior of the polynomial as � → +∞ and � →
−∞? Explain your answer.
(c) Describe, in your own words, the rational zero theorem.
Taking into account the form of the
polynomial, can we use the rational zero theorem to find if the
polynomial has rational
zeros? If yes, what are the possible zeros?

2. (d) Describe in your own words Descartes’ rule of signs. Using
this rule, what is the possible
number of positive zeros of the polynomial, and what is the
possible number of negative
zeros?
(e) Calculate the following values:
�(−3) = ________ �(−2) = ________ �(−1) = ________ �(0) =
________
�(1) = ________ �(2) = ________ �(3) = ________
�(−4) = ________ �(4) = ________
2
(f) Describe in your own words the Factor Theorem. Using this
theorem show that �(�) =
(� + 3)(� + 1)(� − 1)(� − 3) ⋅ �(�).
(g) Using long division, find �(�). Show all your work.

3. (h) Calculate the following values:
�(−1) = ________ �(1) = ________
(i) Using the Factor Theorem and synthetic division, factor
�(�). Show all your work.
(j) Describe the multiplicity of all the zeros of �(�), and
describe the behavior of the graph of
�(�) at these zeros (i.e., is the graph crossing the �-axis at
these zeros or touches and turns
around?).
3

4. (k) What is the maximum number of turning points of �(�)?
(l) Graph �(�).
4
2. We are given the polynomial �(�) = 4�5 + 4�4 + �3 − 2�2
− 2� + 1.
(a) What is the degree of this polynomial? How many zeros
does it have? Do these zeros have
to be real and/or distinct?
(b) What is the behavior of the polynomial as � → +∞ and � →
−∞? Explain your answer.

5. (c) Describe, in your own words, the rational zero theorem.
Taking into account the form of the
polynomial, can we use the rational zero theorem to find if the
polynomial has rational
zeros? If yes, what are the possible zeros?
(d) Describe in your own words Descartes’ rule of signs. Using
this rule, what is the possible
number of positive zeros of the polynomial, and what is the
possible number of negative
zeros?
(e) Calculate the following values:
�(−2) = ________ �(−1) = ________ �(0) = ________
�(1) = ________ �(2) = ________
(f) Describe in your own words the Intermediate Value
Theorem. Based on the results in (e)
above, and the possible rational zeros described in (c), show
that � −
1
2
is a factor of �(�).
Clearly describe your reasoning.

6. 5
(g) Describe in your own words the Factor Theorem. Using this
theorem show that �(�) =
(� + 1)(2� − 1) ⋅ �(�).
(h) Using long or synthetic division, find �(�). Show all your
work.
(i) Calculate the following values:
�(0) = ________ �(1) = ________

7. (j) Using the Intermediate Value Theorem, and the possible
rational zeros described in (c),
show that � −
1
2
is a factor of �(�). Clearly describe your reasoning.
(k) Using the Factor Theorem and synthetic division, factor
�(�). Show all your work.
6
(l) Describe the multiplicity of all the real zeros of �(�), and
describe the behavior of the graph
of �(�) at these zeros (i.e., is the graph crossing the �-axis at
these zeros or touches and
turns around?).

8. (m) What is the maximum number of turning points of �(�)?
(n) Graph �(�).
7
3. Each of the following questions corresponds to a different
function and concept.
(a) Is the function �(�) =
2
3
�8 − 3.12�6 + 2�1/2 − 2 a polynomial? Clearly explain your
answer.
(b) Is the function �(�) =
�2+�−2
�+1
a polynomial? Clearly explain your answer.

9. (c) Two of the zeros of the polynomial �(�) are �1 = −3 and
�2 = 2. The former has a multiplicity
of 1, while the latter has a multiplicity of 4. Describe the
behavior of the graph of �(�) at each
zero.
(d) Use the intermediate value theorem to find the ranges of
values of � in which the polynomial
�(�) = �3 − 6�2 + � + 5 has its zeros. Give each range in the
form (�, �), where � = � + 1.
(e) Find the � and � intercepts of the polynomial �(�) = 6�3
− �2 + 5.
(f) A polynomial �(�) is of degree � = 4. The value � = 1 is a
solution of the polynomial of
multiplicity 2. The value � = 1 − � is also a solution of the
polynomial. Find �(3).
8

10. (g) (i) Using long division, divide the polynomial �(�) = 3�4
− 8�3 + 4�2 − 5� + 6 by the
polynomial �(�) = 3�2 − 11� + 8. Clearly mark the dividend,
divisor, quotient, and remainder.
(ii) Now, noting that �(�) = (� − 1)(3� − 8), use synthetic
division to divide �(�) by (� − 1),
and thus get �(�) = (� − 1) ⋅ ℎ(�), and then use synthetic
division again to divide ℎ(�) by
(3� − 8). Show that the answer is the same as in part (i).
9
4. We want to graph the rational function �(�) =
3�3−2�2−3�+2
�3+4�2+�−6
≡

11. �(�)
�(�)
.
(a) Determine if the graph has �-axis symmetry and/or �-axis
symmetry. Clearly explain your
answer.
(b) Find the �-intercept (if it exists).
(c) Find the �-intercepts (if they exist) by finding the roots of
�(�) = 0.
(d) Find any vertical asymptote(s) by finding the roots of �(�)
= 0.

12. (e) What is the domain of �(�)?
(f) Are there any points on the graph that are holes?
(g) Find the horizontal asymptote (if it exists). Clearly explain
your answer.
10
(h) Plot a few additional points between and beyond each �-
intercept and vertical asymptote.
Graph �(�).
(i) Are there any points where the graph crosses the horizontal
asymptote? Are there any
points where the graph crosses the vertical asymptotes?

13. 11
5. We want to graph the rational function �(�) =
3�3+19�2+4�−12
�2+5�+6
≡
�(�)
�(�)
.
(a) Determine if the graph has �-axis symmetry and/or �-axis
symmetry. Clearly explain your
answer.
(b) Find the �-intercept (if it exists).
(c) Find the �-intercepts (if they exist) by finding the roots of
�(�) = 0.

14. (d) Find any vertical asymptote(s) by finding the roots of �(�)
= 0.
(e) What is the domain of �(�)?
(f) Are there any points on the graph that are holes?
(g) Find the slant asymptote (if it exists). Clearly explain your
answer.

15. 12
(h) Plot a few additional points between and beyond each �-
intercept and vertical asymptote.
Graph �(�).
(i) Are there any points where the graph crosses the slant
asymptote? Are there any points
where the graph crosses the vertical asymptotes?
13
6. (a) Solve the inequality �3 − �2 − � − 2 > 0. Express the
solution set in interval notation.

16. (b) Solve the rational inequality
�
3�+2
≤ −
2�−1
�−1
. Express the solution set in interval notation.