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CHAPTER 2.2
HIGHER DEGREE
POLY’S
2.2 BEGINS: POLYNOMIAL
Graphs
of Polynomial Functions
Polynomial
functions are continuous.
What
this means to us is that the graphs
of polynomial functions have no breaks,
holes, or gaps.
Also
polynomial are considered smooth!
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CHARACTERISTICS
OF POLYNOMIAL FUNCTIONS.
Graphs
of Polynomial Functions
Polynomial functions
have another characteristic. They
have:
Extrema:
The multiple minimums and maximums of a
function. Global = relative.
IS THIS A POLYNOMIAL? ESTIMATE
THE EXTREMA IF YOU CAN!!!
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CONTINUOUS? ? ESTIMATE THE
EXTREMA IF YOU CAN!!!
POLYNOMIAL? ? IF SO, ESTIMATE
THE EXTREMA IF YOU CAN!!!
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DEGREES OF A POLYNOMIAL
This is the highest exponent of a variable. 4ଵଵଵ 6 ଼଼ 2
This would be a 111th degree polynomial.
If you have ଷ ସ ,you would added the exponents together. 7th degree poly.
We will cover this later in the book.
LEADING COEFFICIENT TEST
As
x moves without bound to the left or to the
right, the graph of the polynomial function f(x)
= ௡ ௡ . . . ଵ ଴ eventually rises or
falls in the following manner:
1. When
n is odd:
a. If the leading coefficient is positive, the graph
falls to the left and rises to the right.
b. If the leading coefficient is negative, the graph
rises to the left and falls to the right.
2. When
n is even:
a. If the leading coefficient is positive, the graph
rises to the left and right.
b. If the leading coefficient is negative, the graph
falls to the left and right.
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A TABLE TO REMEMBER BY…
APPLICATION OF THE LEADING
COEFFICIENT TEST
What is the degree of the polynomial?
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EXAMPLE #2
What is the degree of the polynomial?
ZEROS OF POLYNOMIAL
FUNCTIONS
Let
f be a polynomial function of degree
n. The function f has at most n real zeros.
The graph of f has at most
n
- 1 relative extrema.
Example:
This
How many times
The graph crosses
or touches the
x-axis.
3 ଻ 3 2
has up to ____ extrema!
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AT MOST HOW MANY ZEROS AND EXTREMA?
1)
૞ 2)
૛ ૝
3)
૜ ૡ
OTHER VOCABULARY
zero If – ௞ , 1is a factor of a polynomial,
then is a repeated zero.
Repeated
Multiplicity
The number of times a zero is repeated.
2 ଷ 2 2 2. We have a
repeated zero of 2, and it’s multiplicity is 3.
Example:
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WHAT DOES THIS ALL MEAN?
If
a polynomial function f has a repeated
zero x = 3 with multiplicity 4, the graph of f
touches the x-axis at x = 3. (multiplicity is
even, it touches)
If
f has a repeated zero x = 4 with
multiplicity 3, the graph of f crosses the xaxis at x = 4 . (multiplicity is odd, it crosses).
[Note: Sometimes there is a little “wiggle” in
the graph for this situation]
ZEROS OF POLYNOMIAL
FUNCTIONS
Let
f be a polynomial function and let a be a real
number. Four equivalent statements about the
real zeros of f:
1)
is a zero of the function 2)
is a solution of the polynomial equation
0
3)
is a factor of the polynomial 4)
, 0is an x-intercept of the graph of 8
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EXAMPLE 3: FIND THE X INTERCEPTS
Find
the x-intercept of the graph of ଷ – ଶ – 1.
Grouping
method:
ଷ – ଶ – 0
0
0
1.
Note
that in the above example, 1 is a repeated zero. In
general, a factor – ௞ , 1, yields a repeated zero of multiplicity k. If k is odd, the graph crosses the x-axis at
x = a. If k is even, the graph only touches the x-axis at x = a.
YOU TRY THE GROUPING METHOD
1)
૜ ૛ 9
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EXAMPLE 4: CAN YOU SKETCH THE GRAPH?
ଵ
ସ
ସ 2 ଶ 3
Odd
or even value of n?
If
the leading coefficient is positive, the graph rises to the
left and right.
Find
it’s zeros or intercepts.
Stumped?
0
What if I multiply it all by 4?
0
What does this kind of look like?
0
What if I subs. a for ଶ ?
0
Factor it now!
0
Put back the ଶ .
0
Now solve for each.
EXAMPLE 4 CONTINUED.
ଵ
ସ ସ 2 ଶ 3
0
0
0
With
all of the information, we can surely graph to some
degree of certainty. Rises to the left, and right. No
multiplicity/repeated zeros.
!" #
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FIND THE ZEROS: YOU TRY THE FACTORING
METHOD
1)
૝ ૛ $
EXAMPLE #5
Sketch
the graph of ଷ – 2 ଶ .
1.
Since the x-intercepts are , and
, .
2.
The graph will go _____ to
right and ______ to the left.
3.
Additional points on the
graph are
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FINDING ZEROS #6,7
FIND A “POSSIBLE” POLY WITH
GIVEN ZEROES. EXAMPLE 8
Find
the polynomial function with the
following zeros. , , ,
Why
do you think they call this a possible
polynomial?
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FIND THE POLYNOMIAL WITH THE POSSIBLE
ZEROS (OYO)
1)
, $, FIND THE POLYNOMIAL WITH THE POSSIBLE
ZEROS (OYO)
1)
% , &
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EXAMPLE 9: HARDER ONE. LETS
DO THIS! (DOUBLE FOIL)
0
H.W.
P.109 #29-31,37,43-47ODD,49, 53,
55, 59,67,70,71,73, 81, 89,
[CHALLENGE 109].
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