Investigation: Classifying Polynomials
All equations and expressions in math have names so we can identify them, just
like we have names, a first name and a last name.
Part 1: What is a Monomial?
A monomial is a term that can be written as axn, however there are certain
cases that are not monomials.
The following is a list of examples of monomials vs. not monomials
Example
-5x6/2
4x-3
3.2x
-x3/2
7x3
38x-3.6
-6
3 2
x
4
Not
Monomial
Monomial
Reason (for not monomial)










Step 1: What is the difference between the two columns? What kind of
coefficient can a monomial have? What kind of exponent on the variable can a
monomial have?
Step 2: Complete the rest of the chart above.
Part 2: What is a Polynomial?
A polynomial is a monomial or a sum or difference of monomials.
The following is a list of examples of polynomials vs. not polynomials
Example
Poly

-5x2
-2
-16t

4x3 + 0.75x2 – 4x + 2
3
2
-1/6
-3x – 2x + 0.7x + 5x

xy 2

3
Not


Leading
Coefficient
Classification

x2
y4
a2 + 2ab – b20
a-2 + 2ab – b-20

x  26x 2
x4  3 x

8
-10


4
0.44-2 pq


1
pq  qr  rp
0.44 – 2-10p + q4
(x5)(5x-3)(9)
y
 y3
3





Step 1: Study the list. Discuss why each of the examples is on their respective
sides of the list. Then come up with your own example of a polynomial and not a
polynomial.
Step 2: The leading coefficient is the coefficient of the first term when the
polynomial is written in standard form. State the leading coefficients for the
examples in the chart above that are polynomials.
Part 3: Classifications – How to give a polynomial a name!!
Step 1: Classifying by Degree
Degree of a term is the power on the variable of that term
The degree of a polynomial is based of the term with the highest power on a
variable. In standard form, we look to the first term.
Degree
0
1
2
3
4
5
>6
Name
Constant
Linear
Quadratic
Cubic
Quartic
Quintic
___th degree
Example
__________
__________
__________
__________
__________
__________
__________
Leading coefficient
__________
__________
__________
__________
__________
__________
__________
Step 2: Classifying by Number of Terms
Number of terms
1
2
3
>4
Classification
Monomial
Binomial
Trinomial
Polynomial with __
number of terms
Example
__________
__________
__________
__________
Step 3: Now go back to part 2 and classify each of the polynomials by number
of terms and degree.
Step 4: For each of the following, simplify and classify each of the following
polynomials by degree and number of terms, and state the leading coefficient.
Make sure each polynomial is written in standard form.
a) (x3 + 4x2 – 2x + 7) + (3x4 –x3 + 2x2 - 2x + 5)
b) (x2 – 4x – 2) – (3x – 4) + (-x2 – 2x)
c) (2r – 2)(-r – 7)
d) (2x3 +2x2 -2x) – (2x3 +7) + (-2x2 + 2x – 7)
e) 6(8m3 + 9)
f) 8m3(4m2)
g) (2 + v)2
h) (7r5 – 6r – 6)(2r2 – 4)
i) 3a + 4a
j) (12x7 + 4x4 – 2x2 – x + 7) – (3x4 + x3 – 5x2 – 3x + 12)