MATH 4H
POLYNOMIAL FUNCTIONS
HOMEWORK
NAME__________________________
DATE___________________________
Homework assignments are from the Merrill textbook.
HW # 15:
1. Text p. 141 - # 12, 27, 28
2. Text pp. 144-5 - # 2, 26, 35 and for #20 do a complete analysis and graph
3. Packet p. 2 – Polynomial Functions AP Style - #MC1
HW # 16:
Packet p. 9 - Ditto 4-3 - ALL
HW # 17:
Packet p. 1, p. 5 – Polynomial Functions AP Style - #MC2, MC3, FR1
HW # 18:
Packet pp. 5-6 – Polynomial Functions AP Style - #FR2, FR3
HW # 19:
Packet pp. 10-15 – Properties of Graphs of Polynomial Functions Homework
HW # 20:
Packet pp. 3-4, pp. 7-8 – Polynomial Functions AP Style - #MC4 – MC9, FR4 – FR6
HW # 21:
Packet pp. 16-17 – Polynomial Roots
HW # 22:
p. 176 - # 1 - 13 ODD
- # 17, 19, 21, 23
Study for Test!!!
1
Polynomial Functions AP Style
MC1. The function P(x) has a zero at x = 3 which has a multiplicity of 2. Which of the following
statements MUST BE true about the graph of P(x) at x = 3?
A. Since x = 3 is a zero of P(x), then the graph passes through the x-axis at x = 3
B. Since the multiplicity of x = 3 is 2, then the graph is tangent to the x-axis at
x = 3.
C. Since the multiplicity of x = 3 is 2, then the graph is below the x-axis on both
sides of x = 3.
D. The graph of P(x) passes through the x-axis at x = 3 and changes concavity.
E. None of these
MC2. Find the value of k so that (x + 3) is a factor of H(x) = 3x3 – 2x2 + kx – 3.
A. 34
B. 20
C. –20
D. –34
E. 32
MC3. What can be said about the result of P(1) and the remainder when P(x) = 2x3 – 2x2 – x – 3 is
divided by (x – 1).
A.
B.
C.
D.
E.
P(1) is greater than the remainder.
P(1) is less than the remainder.
P(1) is equal to the remainder.
P(1) cannot be determined.
None of these
{OVER}
2
MC4. Use Descartes Rule of Signs to determine the possible number of positive zeros of
f ( x)  6 x 5  6 x 3  10 x  5 .
A.
B.
C.
D.
E.
4, 2, or 0
3 or 1
2 or 0
1
None of these
MC5. The table of values to the right includes points that lie on the graph of f(x), a
polynomial function. Based on the values in the table, between what two
integers is there guaranteed to be a zero of the function?
A.
B.
C.
D.
E.
–3 and –1
–1 and 1
1 and 3
3 and 5
Both B and C
MC6. Which of the following tables correctly shows the possible combinations of roots of the
polynomial function f(x) = –3x4 + 2x2 + 3x – 2.
A.
C.
B.
D.
{OVER}
3
MC7. What is the maximum number of possible negative real roots that the function
f(x) = –2x4 + 3x3 + 2x2 + 3x – 2 can have?
A.
B.
C.
D.
E.
4
3
2
1
0
MC8. Which of the following statements is/are true about polynomial functions?
I.
The graph of a quadratic function can have two points of tangency to the x – axis.
II.
The function f(x) = 2x3 – 3x2 – 2x + 3 has either 2 or 0 positive real roots.
III.
The roots of a certain cubic function can be x = +3i and x = 3 + 2i.
A.
B.
C.
D.
E.
I only
II only
I and II only
II and III only
I, II, and III
MC9. Which of the following statements is NOT true about the function g(x) = 2x3 + 5x2 – 3x.
A. x = 0 is a root of g(x).
B. The graph of g(x) falls to the left and rises to the right.
C. The range of g(x) is (, ).
D. The multiplicity of each root of g(x) is 1.
E. The maximum number of times that the graph of g(x) could cross the x – axis
on the positive side of the origin is 2.
{OVER}
4
FR1.
Free Response #1
Calculator NOT Permitted
Consider the polynomial function f(x) = x3 +3x2 – 25x – 75 to answer the following questions.
a. Rewrite the function in completely factored form by factoring the polynomial by grouping.
b. Use synthetic division to show that (x + 3) is a factor of the function. Explain why your work shows
that it is a factor.
c. Based on your result from part a), can anything be determined about the graph of f(x) at x = –3?
Explain your reasoning.
d. Suppose that g(x) = – 3x3 + kx – 2. For what value of k would (x + 2) be a factor of g(x). Show
your work.
FR2.
Free Response #2
Calculator NOT Permitted
x
–3
–2
0
1
3
4
F(x)
50
16
–4
–2
–4
–20
The table above shows function values of a cubic polynomial function, F(x). The function has two
distinct zeros, x = a and x = b, such that a < 0 and b > 0. Additionally, one of the zeros has a
multiplicity of two.
a. Between what two x – values in the table does the zero x = a lie? What is its multiplicity? Justify
your reasoning.
b. Between what two x – values in the table does the zero x = b lie? What is its multiplicity? Justify
your reasoning.
c. Determine the left and right hand behavior of F(x) based on the table of values. Give a reason for
your answers.
d. What can be said about the leading coefficient of F(x)? Justify your reasoning.
{OVER}
5
FR3.
Free Response #3
Calculator NOT Permitted
Consider the following polynomial function f(x) = –2x3 – x2 + 13x – 6 to answer the following
questions.
a. If (x + 3) is a factor of f(x) rewrite f(x) in completely factored form.
b. State the zeros of the graph of f(x) and their multiplicities.
c. Suppose a cubic polynomial function with a positive leading coefficient, g(x), is such that x = –3 is
a root of multiplicity 1 and x = 2 is a root of multiplicity 2. Sketch a possible graph of g(x). Explain
how you developed your graph.
d. Write the equation of g(x) in standard form. Show your work.
{OVER}
6
FR4.
Free Response #4
Calculator Permitted
Consider the function h(x) = 2x4 – 11x3 – 9x2 + 18x to answer the following questions.
a. Find the values of h(–2) and h(–1). Based on these values, what can be concluded about one of
the zeros of h(x)?
b. Use Descartes’ Rule of Signs to determine the number of possible positive, negative, zero,
and imaginary roots. Make a chart that summarizes your results and determine which
combination from the chart is the correct combination. Explain how you know which one is the
correct one.
c. Show, using synthetic division, that (x – 1) and (2x + 3) are factors of h(x). Explain how you know
based on these results that these binomials are factors.
FR5.
Free Response #5
Calculator NOT Permitted
Answer the following questions about polynomial functions. In each question, you are to provide an
original example, either in graphical or analytical form, of a polynomial function that fits the described
characteristics and describe why it satisfies the requirements. If it is impossible to provide such an
example, give explanation as to why.
a. Sketch a graph of a quartic function that has two negative real roots and one positive root of
multiplicity of 2. Give a graphical justification.
b. Sketch a graph of a cubic function that has one negative real root, one positive real root and one
imaginary root. Give a graphical justification.
c. In analytical form, give an example of a polynomial function whose graph rises as x – values
approach  and falls as x – values approach . Give an analytical justification.
d. In analytical form, give an example of a quartic function that has a maximum of 3 positive roots but
has a maximum of only 1 negative root. Give an analytical justification.
{OVER}
7
FR6.
Free Response #6
Calculator NOT Permitted
Pictured below are graphs of two different polynomial functions. All of the roots of each function are
real—none are imaginary. Answer the questions that follow about the two graphs, Graph A and
Graph B.
GRAPH A
GRAPH B
a. Based on the graph, what type of polynomial function is graphed in Graph A? Explain your
reasoning.
b. Based on the graph, what type of polynomial function is graphed in Graph B? Explain your
reasoning.
c. Identify the left and right hand behavior of the Graph A. Based on this and the type of function that
it is, what can be said about the leading coefficient of the equation of the polynomial function?
Explain your reasoning.
d. Identify the left and right hand behavior of the Graph B. Based on this and the type of function that
it is, what can be said about the leading coefficient of the equation of the polynomial function?
Explain your reasoning.
8
4-3 Practice Worksheet
The Remainder and Factor Theorems
Divide using synthetic division.
1. (𝑥 2 − 5𝑥 − 12) ÷ (𝑥 + 3)
2. (3𝑥 2 + 4𝑥 − 12) ÷ (𝑥 − 5)
3. (2𝑥 3 + 3𝑥 2 − 8𝑥 + 3) ÷ (𝑥 + 3)
4. (𝑥 4 − 3𝑥 2 + 1) ÷ (𝑥 − 1)
Find the remainder for each division. Is the divisor a factor of the polynomial?
5. (2𝑥 3 − 3𝑥 2 − 10𝑥 + 3) ÷ (𝑥 − 3)
6.(2𝑥 4 + 4𝑥 3 − 𝑥 2 + 9) ÷ (𝑥 + 2)
2
7. (10𝑥 3 − 𝑥 2 + 8𝑥 + 29) ÷ (𝑥 + )
8. (2𝑥 4 + 14𝑥 3 − 2𝑥 2 − 14𝑥 ) ÷ (𝑥 + 7)
5
Use the remainder theorem to find the remainder for each division. State whether
the binomial is a factor of the polynomial.
9. (3𝑥 3 − 2𝑥 2 + 𝑥 − 4) ÷ (𝑥 − 2)
10. (𝑥 4 − 𝑥 3 − 10𝑥 2 + 4𝑥 + 24) ÷ (𝑥 + 2)
11. (𝑥 4 + 5𝑥 3 − 14𝑥 2 ) ÷ (𝑥 + 7)
12. (𝑥 3 + 𝑥 2 − 10) ÷ (𝑥 + 3)
Find the value of k so that each remainder is zero.
13. (2𝑥 3 + 𝑘𝑥 2 + 7𝑥 − 3) ÷ (𝑥 − 3)
14. (𝑥 3 + 9𝑥 2 + 𝑘𝑥 − 12) ÷ (𝑥 + 4)
15. Determine how many times 2 is a root of 𝑥 3 − 7𝑥 2 + 16𝑥 − 12=0.
9
Properties of Graphs of Polynomial Functions Homework
1. Which of the following statements is/are true about polynomial functions.
I.
A cubic function can have roots of 2i and 3 + i.
II.
A cubic function that has a zero of multiplicity of 2 and another zero of multiplicity of 1
has three different x-intercepts.
III. A function that has a degree of 4 is called a quartic function.
A.
B.
C.
D.
E.
I and III only
II only
II and III only
III only
I, II, and III
2. The zeros of a function, g(x), are x = 3, x = 2 + 3i and x = 2. If the multiplicity of x = 3 is 2,
then what type of function is g(x)?
A.
B.
C.
D.
E.
quartic
quadratic
cubic
quintic
linear
3. Which of the following statements is/are true about the function g(x) = x3 – 7x2 + 10x.
A.
B.
C.
D.
E.
I.
The graph of g(x) has a relative maximum at (0.880, 4.061).
II.
The zeros of g(x) are x = 2, x = 0, and x = 5
III.
The graph of g(x) has a point of inflection at (2.333, –2.074).
I only
II only
I and II only
II and III only
I, II and III
{OVER}
10
4. The graph of a polynomial function, f(x) is graphed to the right. Which of the following statements
is/are true if f(x) has no imaginary roots?
I.
The degree of f(x) is odd.
II.
f(x) is a quartic function.
III.
If a is the leading coefficient of f(x), then a > 0.
A.
B.
C.
D.
E.
I only
II only
I and III only
II and III only
III only
5. Write a polynomial of lowest degree with roots −1, 1 − 𝑖, and 2 + √3.
{OVER}
11
6.
Domain:
Range:
Intervals of x – values where
f(x) > 0:
Intervals of x – values where
f(x) < 0:
Left End Behavior:
Right End Behavior:
Zeros and their multiplicity:
Type of Function:
Possible Equation:
Intervals of x – values where
f(x) is increasing:
Intervals of x – values where
f(x) is decreasing:
Relative Maximum(s):
Relative Minimum(s):
Absolute Maximum(s):
Absolute Minimum(s):
Point(s) of Inflection:
Intervals of x – values where
f(x) is concave up:
Intervals of x – values where
f(x) is concave down:
{OVER}
12
7.
Free Response #7
Calculator Not Permitted
Pictured above is the graph of a polynomial function, f(x), whose roots are all rational. Use the graph
to answer the questions that follow.
(a) What type of function is f(x) and what can be concluded about the leading coefficient of the
equation? Justify your answers.
(b) Identify any relative maximums and/or minimums of f(x). Then, state the intervals on which f(x)
is increasing and/or decreasing.
(c) Identify the coordinates of any points of inflection. Then, state the intervals on which f(x) is
concave up and/or down.
{OVER}
13
8.
Free Response #8
Calculator NOT Permitted
The table below shows function values and graphical properties for a cubic polynomial function, h(x),
at indicated values or intervals of x.
x
h(x)
(, –3)
Increasing
&
Concave
Down
–3
(–3, –1)
3
Decreasing
& Concave
Down
–1
(–1, 1)
1
Decreasing
& Concave
Up
1
(1, )
–1
Increasing
&
Concave
Up
(a) At what x – value(s) does the graph of h(x) reach a relative maximum? At what x – value(s) does
the graph of h(x) reach a relative minimum? Justify your answer.
(b) Is either of the two relative extrema that you mentioned in parts a and b an absolute extremum?
Justify your answer.
(c) At what x – value(s) does the graph of h(x) have a point of inflection? Justify your answer.
(d) Sketch a possible graph of h(x).
{OVER}
14
9.
Free Response #9
Calculator Permitted
A quartic function, ℎ(𝑥), has zeros of 𝑥 = – 2, 𝑥 = 3, and x = 3  2i .
(a) State the multiplicity of the two real roots, 𝑥 = – 2 and 𝑥 = 3. Explain your reasoning.
(b) Find the equation of ℎ(𝑥).
(c) Draw a sketch of the graph of the function ℎ(𝑥). Based on the graph, how do you know if your
answer in part (a) is correct or not? Completely explain your reasoning.
(d) We know that 𝑥 = – 2 is a zero of ℎ(𝑥). Find ℎ(– 3) and ℎ(1) and explain how these values
guarantee that there is a zero located on the interval (– 3, 1).
15
Polynomial Roots
For each given polynomial:
(a) find the number of possible positive roots.
(b) find the number of possible negative roots.
(c) find the possible rational roots.
(d) find all zeros of the polynomial.
(e) analyze and graph the polynomial.
1. 𝑃(𝑥 ) = 2𝑥 3 − 9𝑥 2 + 6𝑥 − 1
{OVER}
16
2. 𝑃(𝑥 ) = 2𝑥 4 + 5𝑥 3 + 4𝑥 2 + 5𝑥 + 2
17