NAME: _____________________________DATE: ___________________
PERIOD: _____
Seat #: ______
INTRODUCTION TO POLYNOMIALS
A polynomial is a monomial or a sum of monomials. A monomial is an expression with a real number, a variable or a
product of a real number and one or more variables with the following restrictions:



Variables must have whole number exponents.
Variables do not appear under simplified radical expressions.
Denominators to not contain variables.
The degree of a monomial is the sum of the exponents of the variables. The following express ions are examples of
monomials with their degrees:
NOTE:
can be rewritten as
or as
The following express ions are not monomials:
(The variable has an exponent that isn’t a whole number)
(A variable is under the radical sign)
(The denominator has a variable)
Two or more monomials with identical variable expressions are called LIKE TERMS or similar terms. In order to add
monomials, they must be like terms. Unlike terms cannot be added together. To add like terms: (1) Add their numerical
coefficients. (2) Keep the variable expressions.

The following are like terms, since their variable expressions are all

These are not like terms since their variable expressions are not all the same:
:
.
A polynomial is a variable expression in which each term is monomial. The degree of a polynomial is the greatest degree
of any of its terms. For example:



(degree 1 because
)
(degree 2 because
is the term with the greatest degree )
(degree 5 because the exponents of the 1st term,
are 3+2=5)
A polynomial in x is defined as a finite sum of terms of the form
, where a, a real number, is the leading coefficient
(coefficient of term with LARGEST exponent, and the exponent n is called the degree of the term. For example:
Term (Expressed in
the form
)
Coefficient
 rewrite as
 rewrite as
Degree
7
 rewrite as
Give the coefficient and the degree of each term:
Term (Expressed in
the form
)
Coefficient
Degree
1.
2.
The terms of a polynomial are usually arranged so that the exponents of the variables, called the degree, decrease from
left to right. This is called descending order. In descending order, the highest-degree term is written first and is called
the leading term. Its coefficient is called the leading coefficient. The degree of a polynomial is the greatest degree of all
its terms. Thus, the leading term determines the degree of a polynomial.
Expression
Descending Order
Monomials 2x 9
-49
Binomials
2x9
-49
10y - 7y2
-7y2
10 - 0.67b
+ 10y
- 0.67b+10
Trinomials w + 2w3+ 9w6
9w6 + 2w3 + w
4
8
3
2.5a - a + 1.3a -a8 + 2.5a4 + 1.3a3
Leading
Coefficient
Degree of
Polynomial
2
9
-49
0
-7
2
1
-0.67
9
-1
6
8
-
Polynomials may have more than one variable. In such a case, the degree of the term is the sum of the
exponents of the variables contained in the term. For example, the term
has degree 8 because the
exponents applied to x, y, and z are 3, 4, and 1, respectively. The following polynomial has a degree of 12
because the highest degree of its terms is .
Degree 8
Degree 12
Degree 3
Degree 0
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Write the polynomial in descending order (called Standard Form); then provide the leading coefficient & the degree.
Expression
Descending Order
Leading
Coefficient
Degree of
Polynomial
3.
4.
5.
6.
The degree of a polynomial function (n) affects the shape of its graph and …
1. Determines the number of real and imaginary solutions (n=the number of solutions)
2. Determines the maximum number of turning points, or places where the graph changes direction.
 The maximum number of turning points is (
) where n is the degree of the polynomial (recall that the term with
the highest exponent determines the degree).
o If a polynomial has degree 4 there will be a maximum of 3 turning points.
3. Affects the END BEHAVIOR or the directions of the graph to the far left and to the far right.
 You can determine the end behavior of a polynomial function of degree n from the leading term
of the standard
form.
A function is increasing (rising) when “a” is positive and the y-values increase as the x-values increase.
A function is decreasing (falling) when “a” is negative the y-values decrease as the x-values increase.
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Example Problem 1: Write the following polynomials in standard form. What is the classification of each by degree
(Name Using Degree)? By the number of terms (Name Using Number of Terms)?
Expression
Standard Form
(combine like terms)
Name Using
Degree
Name Using
Number of
Terms
7.
8.
9.
Example Problem 2: What is the classification of the following polynomial by its degree? by its number of
terms? What is its end behavior?
5x4 − 3x + 4x6 + 9x3 − 12 − x6 + 3x4
Step 1
Write the polynomial in standard form. First, combine any like terms. Then, place the terms
of the polynomial in descending order from greatest exponent value to least exponent value.
5x4 − 3x + 4x6 + 9x3 − 12 − x6 + 3x4
Combine like terms.
Place terms in descending order.
Step 2
The degree of the polynomial is equal to the value of the greatest exponent. This will be the
exponent of the first term when the polynomial is written in standard form.
3x6 + 8x4 + 9x3 – 3x – 12
The first term is _______.
The exponent of the first term is______.
This is a ________-degree polynomial.
Step 3
Count the number of terms in the simplified polynomial. It has___ terms.
Step 4
To determine the end behavior of the polynomial (the directions of the graph to the far left and
to the far right), look at the degree of the polynomial (n) and the coefficient of the leading term
(a). If a is positive, it is rising; if a is negative it is falling.
If a is positive and n is even, the end behavior is up and up.
If a is positive and n is odd, the end behavior is down and up.
If a is negative and n is even, the end behavior is down and down.
If a is negative and n is odd, the end behavior is up and down.
The leading term in this polynomial is______
a (___) is _______ and n (___) is even, so the end behavior is ____________.
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Example Problem 3: Describe the graphs of:
a)
b)
Example Problem 4: What is the degree of the polynomial function that generates the data shown below? What are the
differences when they are constant? To find the degree of a polynomial function from a data table, you can use the
differences of the y-values.
Step 1 Determine the values of y2 – y1, y3 – y2, y4 – y3, y5 – y4, y6 – y5, y7 – y6. These are called the first differences. Make
a new column using these values.
Step 2 Continue determining differences until the y-values are all equal. The quantity of differences is the degree of the
polynomial function. The ____ differences are all equal so this is a _____ degree polynomial function. The
value of the _______ differences is ____.
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