math 71
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Math 71
5.1 – Introduction to Polynomials and Polynomial Functions
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A number or a number multiplied by variables is called a ________________(also called a ________).
Polynomials
3
A number or a number multiplied by variables is called a ________________(also called a ________).
Polynomials
monomial
4
A number or a number multiplied by variables is called a ________________(also called a ________).
Polynomials
monomialterm
5
Ex 1.Which of the following are monomials? 17
Polynomials
6
Polynomials
Ex 1.Which of the following are monomials? 17
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The sum of the exponents of the variables is called the ________________ of a monomial.
The numerical factor in a monomial is called the ___________________.
Polynomials
8
The sum of the exponents of the variables is called the ________________ of a monomial.
The numerical factor in a monomial is called the ___________________.
Polynomials
degree
9
The sum of the exponents of the variables is called the ________________ of a monomial.
The numerical factor in a monomial is called the ___________________.
Polynomials
degree
coefficient
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Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
17
11
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
17
12
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
13 5
17
13
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
13 5
4 2
17
14
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
13 5
4 2
1
17
15
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
13 5
4 2
1
2 1
17
16
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
13 5
4 2
1
2 1
1 -1
17
17
Ex 2.What are the degrees and coefficients of the following monomials?
PolynomialsMonomial Degree Coefficient
2 -4
13 5
4 2
1
2 1
1 -1
17 0 17
18
One or more monomials added together is called a ___________________.
ex:
The highest degree of all terms in a polynomial is called the _________________ of the polynomial.
ex: What is degree of above polynomial?
Polynomials
19
One or more monomials added together is called a ___________________.
ex:
The highest degree of all terms in a polynomial is called the _________________ of the polynomial.
ex: What is degree of above polynomial?
Polynomials
polynomial
20
One or more monomials added together is called a ___________________.
ex:
The highest degree of all terms in a polynomial is called the _________________ of the polynomial.
ex: What is degree of above polynomial?
Polynomials
polynomial
degree
21
One or more monomials added together is called a ___________________.
ex:
The highest degree of all terms in a polynomial is called the _________________ of the polynomial.
ex: What is degree of above polynomial?
Polynomials
polynomial
degree
2+3=𝟓
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A polynomial with 2 terms is called a ____________________.
A polynomial with 3 terms is called a ____________________.
Polynomials
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A polynomial with 2 terms is called a ____________________.
A polynomial with 3 terms is called a ____________________.
Polynomials
binomial
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A polynomial with 2 terms is called a ____________________.
A polynomial with 3 terms is called a ____________________.
Polynomials
binomial
trinomial
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Ex 3.Add:
Adding Polynomials
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Ex 3.Add:
Adding Polynomials
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Ex 4.Subtract:
Subtracting Polynomials
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Ex 4.Subtract:
Subtracting Polynomials
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Ex 4.Subtract:
Subtracting Polynomials
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Here’s a polynomial function: And here’s what looks like:
Polynomial Functions
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In general, polynomial functions are __________ (i.e.
only have rounded curves with no sharp corners) and
________________ (i.e. don’t have breaks and can be
drawn without lifting pencil).
Polynomial Functions
32
In general, polynomial functions are __________ (i.e.
only have rounded curves with no sharp corners) and
________________ (i.e. don’t have breaks and can be
drawn without lifting pencil).
Polynomial Functions
smooth
33
In general, polynomial functions are __________ (i.e.
only have rounded curves with no sharp corners) and
________________ (i.e. don’t have breaks and can be
drawn without lifting pencil).
Polynomial Functions
smooth
continuous
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The following are not the graphs of polynomial functions:
Polynomial Functions
35
The following are not the graphs of polynomial functions:
Polynomial Functions
Not smooth(Sharp corner—ow!)
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The following are not the graphs of polynomial functions:
Polynomial Functions
Not continuousat
Not smooth(Sharp corner—ow!)
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The question “What is the end behavior of a function?” means “What does the function do as it goes to the far left or far right?”
That is, does it go up? Go down? Wiggle up and down? Get flatter?
End Behavior
38
The question “What is the end behavior of a function?” means “What does the function do as it goes to the far left or far right?”
That is, does it go up? Go down? Wiggle up and down?
End Behavior
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For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree).
Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values…
End Behavior
40
For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree).
Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values…
End Behavior
leading term
41
For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree).
Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values…
End Behavior
leading termhighest
42
For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree).
Why? Let’s look at what happens to the polynomial function when we plug in bigger and bigger positive -values…
End Behavior
leading termhighest
43
For polynomial functions, end behavior depends on the ___________________ (the term with _______________ degree).
Why? Let’s look at what happens to the polynomial function when we plug in big positive -values…
End Behavior
leading termhighest
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1 -2 10 1897 100 1989997 1000 1998999997 3
The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.
End Behavior
45
1 -2 10 1897 100 1989997 1000 1998999997 3
The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.
End Behavior
46
1 -2 10 1897 100 1989997 1000 1998999997 3
The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.
End Behavior
47
1 -2 10 1897 100 1989997 1000 1998999997 3
The term grows faster than or . So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.
End Behavior
48
1 -2 10 1897 100 1989997 1000 1998999997 3
The term grows faster than the and terms. So, since becomes a bigger and bigger positive # as becomes a bigger and bigger positive #, the graph of rises as it goes to the right.
End Behavior
49
So, to determine the end behavior of a polynomial function, think about what would happen if you plugged in a really big negative -value and a really big positive -value into the leading term.
(In fact, any negative # and positive # will do.)
50
Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
51
Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟑 𝒙𝟔
52
Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟑 𝒙𝟔
is positive
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Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟑 𝒙𝟔
rises
is positive
54
Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟑 𝒙𝟔
rises
is positive is positive
55
Ex 5.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟑 𝒙𝟔
rises rises
is positive is positive
56
risesrises
57
Ex 6.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
58
Ex 6.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
−𝟑 𝒙𝟒
59
Ex 6.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
−𝟑 𝒙𝟒
is negative
60
Ex 6.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
−𝟑 𝒙𝟒
falls
is negative
61
Ex 6.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
−𝟑 𝒙𝟒
falls
is negative is negative
62
Ex 6.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
−𝟑 𝒙𝟒
falls falls
is negative is negative
63
fallsfalls
64
Ex 7.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
65
Ex 7.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟐 𝒙𝟓
66
Ex 7.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟐 𝒙𝟓
is negative
67
Ex 7.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟐 𝒙𝟓
falls
is negative
68
Ex 7.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟐 𝒙𝟓
falls
is negative is positive
69
Ex 7.Find the leading term, and determine the end behavior of the graph of Leading term: ________.
Graph ________ to left and ________ to right.
𝟐 𝒙𝟓
falls rises
is negative is positive
70
rises
falls
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