Zeros of Polynomial Functions
Numerical, Analytical and Graphical Approaches
For each of the functions graphed below, state the end behavior and the zeros of the functions.
Graph of f(x)
Graph of g(x)
Graph of h(x)
f(x) is a ________ function.
g(x) is a _________ function.
h(x) is a _________ function.
Left End Behavior
Left End Behavior
Left End Behavior
Right End Behavior
Right End Behavior
Right End Behavior
Zeros
Zeros
Zeros
Define the multiplicity of a zero.
Read the following information about the multiplicities of the zeros of f(x), g(x), and h(x) while studying
the graphs above. Then, answer the questions on the next page.
In the graph of f(x), all of the zeros have a multiplicity of 1.
In the graph of g(x), the zero of x = –2 has a multiplicity of 1 and x = 2 has a multiplicity of 3.
In the graph of h(x), the zeros x = –4, x = –2, and x = 5 have a multiplicity of 1 and x = 2 has a
multiplicity of 2.
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1. What do you notice about the sum of the
multiplicities of the zeros and the degree of
the
function?
2. Describe the behavior of the graph as it
approaches a zero whose multiplicity is 1.
3. Describe the behavior of the graph as it
approaches a zero whose multiplicity is 2.
4. Describe the behavior of the graph as it
approaches a zero whose multiplicity is 3.
Examples:
(a) f(x) = x3 + 2x2 – x – 2
Type of function:________________________________________
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
(b) h(x) = 2x3 – x2 – 4x + 3
Type of function:________________________________________
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
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(c) p(x) = 2x4 – 7x3 – 6x2 + 44x – 40
Type of function:________________________________________
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
Root:________________ Multiplicity:__________________
Describe the behavior of the graph at this root.
Now, let’s consider how we might be able to locate the zeros of a polynomial function numerically.
Consider the function h(x) = 2x3 – x2 – 4x + 3 that we investigated earlier and whose graph is shown
below. Find each pair of function values in the table below and answer the questions that follow
Find h(–2) and h(–1).
Find h(0) and h(2).
From the graph, clearly h(x) has a zero between x = –2 and x = –1. Explain how your finding the
values of h(–2) and h(–1) above numerically shows that there is a zero that exists between x = –2 and
x = –1.
Does the same reasoning that you described concerning the zero between x = –2 and x = –1 hold
true for the existence of a zero between x = 0 and x = 2? Explain your reasoning.
Based on what we have just seen, what inference can you make about the existence of a zero of a
polynomial function if you know the value of the function at two different x – values?
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Ex.: The table of values below represents values of a cubic function. The function has a negative
zero and two positive zeros. Answer the questions that follow.
x
−2
−1
0
1
2
4
5
F(x)
−45
−8
3
0
−5
27
88
(a) Is/Are any of the zeros of F(x) specifically identified in the table? Explain your reasoning.
(b) Between which two x – values in the table is the negative zero located? Explain your
reasoning.
(c) Between which two x – values in the table is the second positive zero located? Explain
your reasoning.
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The Remainder Theorem of Polynomial Functions
I. Divide 12,372 by 11. Then, identify the divisor, the dividend, the quotient, and the remainder.
Divisor: _
_______ Dividend: _
Quotient: _
__________
_________________
Remainder: _
___________
Answer to the division problem:____________________________
II. (a) Divide
by
.
Method 1: Long Division
(b) Divisor: _
Quotient: _
Method 2:
____________
Dividend: _
_________________
Remainder: _
Answer to the division problem:____________________________
(c) Division Algorithm:
5
_______________________
__________________
(5 x
1.
4
+ 7 x − 10 x + 4 x − 3)
÷ (x − 1)
3
2
(x
2.
5
− 3 x − 2 x + 1) ÷ (x − 4 )
2
(2 x
3.
3
+ x − 13x + 6)
÷ (2 x − 1)
2
Perform synthetic division
Perform synthetic division
Perform synthetic division
Divisor
Divisor
Divisor
Dividend
Dividend
Dividend
Quotient
Quotient
Quotient
Remainder
Remainder
Remainder
Answer to the division
Answer to the division
Answer to the division
_____
_____
_____
From the five previous examples you should realize that there is a relationship between the
remainder when a polynomial, P(x), is divided by a linear binomial, (x – a), and the value of the
function P(a). What is the relationship?
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Remainder Theorem
Complete the following statement.
If a polynomial function, f(x), is divided by a factor, (x – a), then the remainder will be the same
value as __________________.
1. Find the remainder when
is divided by
2. Find the remainder when
.
is divided by
.
3. For what value of k will the function P(x) = –2x3 – 2x2 + kx – 2 have a remainder of 8 when divided
by the factor (x + 2)?
4. For what value of k will the function P(x) = 3x3 + kx2 – 5 have a remainder of 4 when divided by
the factor (x – 3)?
5. For what value of k will the function P(x) = –x4 – 2x2 + kx – 6 have a remainder of 0 when divided
by the factor (x + 1)?
6. Use synthetic division to find
and
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if
.
Factor Thm, Fundamental Thm of Algebra, and End Behavior
I. Factor Theorem: The complex # c is a zero of a polynomial
p (x) .
p (x) if and only if x − c is a factor of
[ p (c ) = 0]
1. Is x − 2 a factor of p( x) = 2 x3 − 5 x + 6 ?
2. Determine which of the following factors are factors of the function
and explain your work.
A.
B.
, Show
C.
3. The function in number 2 is a cubic function whose highest exponent is 3. Thus, there should be 3
zeros of the function. Based on your work in number 3, what is the other zero of
? What is the
multiplicity of each root? Explain your reasoning.
4. Find the values of k for which x +
1
is a factor of p( x) = 3x 4 − 2 x3 − 10 x 2 + 3kx + 3.
3
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II. Fundamental Theorem of Algebra: Every polynomial of degree n > 0 has at least one
zero (root), possibly imaginary.
Corollary: A polynomial function of degree n has EXACTLY ________ ZEROS.
Theorem: Every polynomial of degree n > 0 can be written as a product of a constant
and n linear factors.
III. Def.: A function that is symmetric with the __________ is called an ________________________.
A polynomial is an ____________________ if all of the exponents of
are _____________.
Def.: A function that is symmetric with the __________ is called an ________________________.
A polynomial is an ____________________ if all of the exponents of
Ex.: Determine if each polynomial is even, odd, or neither.
1.
2.
3.
4.
5.
9
are _____________.
IV. For each of the functions in the chart below, graph the polynomial function, paying close attention
as to how the function comes into the view screen and exits the view screen of the calculator.
Based on your observations of the function being graphed, fill in the chart below.
Function
1.
f(x) = 2x2 – 3x + 3
2.
f(x) = –x4 + x3 – x2 + 3
3.
f(x) = –x2 – 3x + 4
4.
f(x) = x4 – x3 + 2x + 3
5.
f(x) = x3 – 3x2 + 2x – 1
6.
f(x) = x5 – x3 + 2x + 6
7.
f(x) = –x3 + 2x2 – x + 4
8.
f(x) = –x5 + 4x4 – 3x3 + x2 – 2
9.
f(x) = –2x4 – x3 + 8x2 + 12x
Even
or Odd
Degree
Positive or
Negative
Leading
Coefficient
Rise or Fall
to the Left
Rise or Fall
to the Right
Now, write some conjectures about the end behavior of polynomial functions.
Even degree → Left & Right End Behavior is _________________________________.
Odd degree → Left & Right End Behavior is _________________________________.
1. When
and the degree is even, the graph _________________________________________
_____________________________________________________________________________.
2. When
and the degree is odd, the graph __________________________________________
_____________________________________________________________________________.
3. When
and the degree is even, the graph _________________________________________
_____________________________________________________________________________.
4. When
and the degree is odd, the graph __________________________________________
_____________________________________________________________________________.
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