Naming Polynomials
8.1
Part 1
What is a Polynomial?
Here are some definitions….
Definition of Polynomial
An expression that can have constants, variables
and exponents, but:
* no division by a variable
(can’t have something like )
* a variable's exponents can only be 0,1,2,3,... etc
(exponents can’t be fractions or negative)
* it can't have an infinite number of terms
Here’s another definition
• A polynomial is a mathematical expression
consisting of a sum of terms, each term
including a variable or variables raised to a
power and multiplied by a coefficient.
Polynomials look like this…
•
•
•
•
•
•
•
4x² + 3x – 1
8
9xy²
3x – 2y
x³
25x² - 4
5x³ – 4x + 7
Names of Polynomials
A Polynomial can be named in two ways
• It can be named according to the number of
terms it has
• It can be named by its degree
Names by the number of terms:
1 term : monomial
Here are some monomials…
3x²
7xy
x
8
½x
2 terms : Binomial
Here are some binomials…
5x + 1
3x² - 4
x+y
3 terms : Trinomial
Here are some trinomials…
7x² + 2x – 10
4 or more terms – polynomial
There is no special name for polynomials with
more than 3 terms, so we just refer to them as
polynomials (the prefix “poly” means many )
Examples
1.
2.
3.
4.
Name each expression based on
its number of terms
5x + 1
7x²
5x – 2xy + 3y
6x³ - 9x² + x – 10
1.
2.
3.
4.
5x + 1 Binomial
7x²
Monomial
5x – 2xy + 3y Trinomial
6x³ - 9x² + x – 10 Polynomial
Finding Degrees
In order to name a polynomial by
degree, you need to know what
degree of a polynomial is, right??
Finding Degrees
Definition of Degree
The degree of a monomial is the sum of the
exponents of its variables.
For example,
The degree of 7x³ is 3
The degree of 8y²z³ is 5
The degree of -10xy is 2
The degree of 4 is 0 (since
)
The degree of a polynomial in one
variable is the same as the greatest
exponent.
For example,
The degree of
is 4
The degree of 3x – 4x² + 10 is 2
Examples
Find the degree of each polynomial
1.
2.
3.
4.
5.
7x
x² + 3x – 1
10
9x²y³
12 – 13x³ + 4x + 5x²
1.
2.
3.
4.
5.
7x 1
x² + 3x – 1 2
10 0
9x²y³ 5
12 – 13x³ + 4x + 5x² 3
Names of Polynomials by their Degree
Degree of 0 : Constant
For example,
7
-10
8
Degree of 1 : Linear
For example,
3x – 2
½x + 7
12x – 1
Degree of 2 : Quadratic
For example,
7x² - 3x + 6
4x² - 1
Degree of 3 : Cubic
For example,
8x³ + 5x +9
2x³ - 11
Anything with a degree of 4 or more does not
have a special name 
Examples
Name each Polynomial by its degree.
1.
2.
3.
4.
5.
10x³ + 2x
3x + 8
6
9x² + 3x – 1
1.
2.
3.
4.
5.
10x³ + 2x
3x + 8
6
9x² + 3x – 1
Cubic
Linear
Constant
Quadratic
Not a polynomial!
Putting it all together…
Examples
Classify each polynomial based on its degree
and the number of terms:
1. 7x³ - 10x
2. 8x – 4
3. 4x² + 11x – 2
4. 10x³ + 7x² + 3x – 5
5. 6
6. 3x² - 4x
1.
2.
3.
4.
5.
6.
7x³ - 10x
8x – 4
4x² + 11x – 2
10x³ + 7x² + 3x – 5
6
3x² - 4x
cubic/binomial
linear/binomial
quadratic/trinomial
cubic/polynomial
constant/monomial
quadratic/binomial
Standard Form
• STANDARD FORM of a polynomial means that
all like terms are combined and the exponents
get smaller from left to right.
Examples
Put in standard form and then name the
polynomial based on its degree and number of
terms.
1. 4 – 6x³ – 2x + 3x²
2. 3x² - 5x³ + 10 – 7x + x² + 4x
1. 4 – 6x³ – 2x + 3x²
= -6x³ + 3x² – 2x + 4
cubic/polynomial
2. 3x² - 5x³ + 10 – 7x + x² + 4x
= -5x³ + 4x² – 3x + 10
cubic/polynomial
Summary
Names by Degree
• Constant
• Linear
• Quadratic
• Cubic
Names by # of Terms
• Monomial
• Binomial
• Trinomial
A word about fractions…
Coefficients and Constants can be fractions.
½x + 5 is ok!
-3x² + ½ is ok!
is not a polynomial
is not a polynomial
Assignment
Page 373
# 1 – 20
Must write problem for credit.
No partial credit if incomplete.
Summary
Polynomial
Degree
Name by
Degree
Number
of Terms
Name by
Terms
Copy the table and fill in the blanks.
7x³ - 2
3
6x² - 10x + 1
4x + 5
Check yourself!
Polynomial
Degree
Name by
Degree
Number
of Terms
Name by
Terms
7x³ - 2
3
Cubic
2
Binomial
3
0
Constant
1
Monomial
6x² - 10x + 1
2
Quadratic
3
Trinomial
4x + 5
1
Linear
2
Binomial