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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
ƒ Polynomial Functions. Definitions and terminology.
Let n be a nonnegative integer and let
an , an −1 , an − 2 , ... , a2 , a1, a0 be real number with
an ≠ 0. The function defined by
f ( x ) = an x n + an −1 x n −1 + ⋅⋅⋅ + a2 x 2 + a1 x + a0
Is called a polynomial function of x of degree n
(in general form).
Degree: n, highest power.
Leading term: an x n , term with highest power.
Leading coefficient: the coefficient of the variable
to the highest power.
Problem #1.
What is the degree, leading term, and leading coefficient of
the polynomial
a) f ( x ) = 3 x 2 − 7 x + 5
b) f ( x ) = −7 x10 − 5 x8 + 3 x 5 − 7 x + 16
c) f ( x ) = 15
ƒ Domain.
ƒ Graphs of the Polynomial Functions are continuous smooth
curves without jumps and sharp corners.
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
ƒ Examples of graphs of Polynomials.
ƒ “End behavior” and Leading Coefficient test.
The behavior of the graph of a function to the far right
( as x →∞ ) ,and to the far left ( as x → −∞ ) is called the end
behavior of a function.
The “end behavior” of any polynomial
function is the same as that of y = an x n , the
term of highest degree (leading term).
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
ƒ Types of the “end behavior” of Polynomials.
ƒ Hands test for “end behavior”.
Odd type.
f ( x) →
f ( x) →
f ( x) →
f ( x) →
as x → − ∞
as x → ∞
as x → − ∞ as x → ∞
Even type
f ( x) →
f ( x) →
f ( x) →
f ( x) →
as x → − ∞
as x → ∞
as x → − ∞
as x → ∞
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
Problem #2.
State the “end behavior” of following polynomial functions.
a) f ( x ) = 3 x 4 − 50 x 3 − 11x 2 + 33 x + 125
f ( x) →
f ( x) →
as x → − ∞
as x → ∞
b) f ( x ) = −15 x 5 + 10 x 4 − 25 x 3 + 22 x 2 + 11x − 111
f ( x) →
f ( x) →
as x → − ∞
as x → ∞
ƒ Range of the Polynomial Functions.
Odd type.
( −∞ , ∞ )
Even type.
Individual
ƒ Turning points of the Polynomial Functions.
The turning points of a polynomial are the points where the
graph changes direction from increasing to decreasing or vice
versa. The function value at a turning point is either a local
minimum of f or a local maximum of f.
In general, if n is the degree of a polynomial function, then
the graph of f has at most n − 1 turning points.
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
ƒ Y-intercept of the graphs of Polynomials.
Question: Does every polynomial function have a
y-intercept? Give the reason for your answer.
Problem #3.
Find the domain, range and y-intercept for the polynomial
a) f ( x ) = −5 x 3 − 11x 2 + 33 x − 75
b) f ( x ) = 2 x 5 − 11x 3 + 15 x 2 − 20 x + 150
c) f ( x ) = 3 x3 ( x − 2 ) ( 2 x − 1)
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ƒ X-intercepts of the graphs of Polynomials.
Problem #4. Find the x-intercepts (zeros) for
a) f ( x ) = x 3 − 2 x 2 − 5 x
b) f ( x ) = 3 x 3 ( x − 2 ) ( 2 x − 1)
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c) f ( x ) = x 4 − 4 x 2 + 4
d) f ( x ) = x 4 − 6 x 3 + 9 x 2
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
ƒ Multiplicity and x-intercepts.
Definition.
If the factor ( x − a ) occurs k times ( k ≥ 2 ) in
the complete factorization of the polynomial
f , then a is called a repeated zero with
multiplicity k.
Problem #5.
Find the degree and state multiplicity for each zero of the
polynomial
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a) f ( x ) = 3 x 3 ( x − 3) ( 2 x − 5 )
b) f ( x ) = − x5 ( x + 3) ( x − 5 )
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ƒ Patterns of x- intercepts and multiplicity of zeros.
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
Problem #6.
The graph of a polynomial function f is shown below.
Use the graph to answer the
following questions.
a) Is the degree of the polynomial
function f even or odd?
b) Is the leading coefficient positive
or negative?
c) The number of real zero(s) of f is _________________
d) Label with letters ( A, B, …) on the graph all turning points of
the polynomial f .
e) Is the multiplicity k of the zero x = 0 odd or even?
f) Is the multiplicity k of x = 6 the zero equal to1 or greater
than 1?
g) State the "End behavior" of f .
f ( x) →
when x → − ∞ ;
f ( x) →
when x → ∞
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
Problem #7.
The graph of a polynomial function f is shown below.
Use the graph to answer the
following questions.
a) Is the degree of the
polynomial function f even
or odd?
b) Is the leading coefficient
positive or negative?
c) The number of real zero(s) of f is _________________
d) Label with letters ( A, B, …) on the graph all turning points of
the polynomial f.
e) Is the multiplicity k of the zero x = −1 odd or even?
f) Is the multiplicity k of the zero x = −1 equal to1 or greater
than 1?
g) State the "End behavior" of f .
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Math 110
Polynomial Functions. CH 3.2 (PART I).
Lecture #14 and #15
ƒ Sketching the graph of a Polynomial Function that satisfied a
given description.
Problem #8.
Sketch the graph of a polynomial function of even degree
which has a negative leading coefficient, x = 5 as a zero
multiplicity 2 and x = 0 as a zero multiplicity 1.
Problem #9.
Sketch the graph of a polynomial function of odd degree
which has a positive leading coefficient, x = −1 s a zero
multiplicity 1 and x = 2 as a zero multiplicity 2 and
y-intercept ( 0, − 6 ) .
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