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Introduction to Polynomials
Learning Targets
Identifying Parts Of A Monomial
I will be able to:
Classify polynomials by the number of terms
Classify Polynomials By Degree
IDENTIFYING PARTS OF A MONOMIAL
Coefficient
Variable
Exponent
Let’s try an example: Identify the coefficient, variable, and exponent:
Coefficient
Variable
Exponent
WAYS TO CLASSIFY POLYNOMIALSWe can classify polynomials by the number of terms:Monomial: 1 term Think about other words with the prefix
mono: monotone, monochromatic, monologue
Binomial: 2 terms Think about other words with the prefix bi: bicycle, bifocals, bimonthly
Trinomial: 3 terms Think about other words with the prefix tri: tricycle, triathlon, triceratops
Polynomial: 4 or more terms Think about other words with the prefix poly: polytheistic, polygon
Let’s take a closer look at classifying polynomials by number of terms...
Polynomials are fun!
CLASSIFYING POLYNOMIALS BY NUMBER OF TERMSMonomial: a number, a variable, or the product of a number and one or more variables. We are also going to call this a term.
Let’s check out some examples of monomials:
A monomial with no variables is called a constant.
CLASSIFYING POLYNOMIALS BY NUMBER OF TERMS
Binomial: a polynomial with 2 terms
Let’s check out some examples of binomials:
Trinomial: a polynomial with 3 terms
Let’s check out some examples of trinomials:
CLASSIFYING POLYNOMIALS BY DEGREE
Finding the degree of a Monomial: The sum of the exponents of its variables.
Example 1:
Finding the degree of a Polynomial: The same as that of its term with the greatest degree.
Example 1:
Example 2:
Example 2:
A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.
The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.
Example 1: Finding the Degree of a Monomial
Find the degree of each monomial.
A. 4p4q3
The degree is 7. Add the exponents of the variables: 4 + 3 = 7.
B. 7ed
C. 3
Check It Out! Example 1
Find the degree of each monomial.
a. 1.5k2m
b. 4x
b. 2c3
CLASSIFYING POLYNOMIALS BY DEGREE
Finding the degree of a Polynomial: The same as that of its term with the greatest degree.
Example 1:
Example 2:
Some polynomials have special names based on their degree and the number of terms they have.
Degree Name
0
1
2
Constant
Linear
Quadratic
3
4
5
6 or more 6th,7th,degree and so on
Cubic
Quartic
Quintic
NameTerms
Monomial
Binomial
Trinomial
Polynomial4 or more
1
2
3
Find the degree of each polynomial.
Example 2: Finding the Degree of a Polynomial
And its name
A. 11x7 + 3x3
11x7: degree 7 3x3: degree 3
The degree of the polynomial is the greatest degree, 7, so it’s 7th.
Find the degree of each term.
B.
The degree of the polynomial is the greatest degree, 4, so it’s quartic.
Check It Out! Example 2
Find the degree and the name of each polynomial.
a. 5x – 6
b. x3y2 + x2y3 – x4 + 2
CLASSIFYING POLYNOMIALS BY DEGREE
Degree Name Example
NON-EXAMPLES OF POLYNOMIALS
Fractions, Division
Square Roots
Variables as the exponent
Negatives as the exponent
Remember...these are NOT polynomials!
The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.
The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.
Write the polynomial in standard form. Then give the leading coefficient.
Example 3A: Writing Polynomials in Standard Form
6x – 7x5 + 4x2 + 9
Find the degree of each term. Then arrange them in descending order:
6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9
Degree 1 5 2 0 5 2 1 0
–7x5 + 4x2 + 6x + 9.The standard form is The leading coefficient is –7.
Write the polynomial in standard form. Then give the leading coefficient.
Example 3B: Writing Polynomials in Standard Form
y2 + y6 − 3y
Check It Out! Example 3a
Write the polynomial in standard form. Give the leading coefficient. Then name it by degree and number of terms.
16 – 4x2 + x5 + 9x3
18y5 – 3y8 + 14y
Check It Out! Example 3b
Write the polynomial in standard form. Give the leading coefficient. Then name it by degree and number of terms.
Classify each polynomial according to its degree and number of terms.
Example 4: Classifying Polynomials
A. 5n3 + 4nDegree 3 Terms 2
5n3 + 4n is a cubic binomial.
B. 4y6 – 5y3 + 2y – 9
C. –2x
Classify each polynomial according to its degree and number of terms.
D. x3 + x2 – x + 2
E. 6
F. –3y8 + 18y5 + 14y
Lesson Closing
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
Write each polynomial in standard form. Then
give the leading coefficient.
3. 24g3 + 10 + 7g5 – g2
4. 14 – x4 + 3x2
4
5
–x4 + 3x2 + 14; –1
7g5 + 24g3 – g2 + 10; 7
Lesson Closing: Part II
Classify each polynomial according to its degree and number of terms.
5. 18x2 – 12x + 5 quadratic trinomial
6. 2x4 – 1 quartic binomial
