After studying this lesson, you will be able to:

• Find the degree of a polynomial.
• Classify polynomials.
• Put polynomials in descending order.

A Polynomial is a monomial or a sum ofmonomials.

There are 3 Special Names of Polynomials :

Monomials: have one term

Binomials: have two terms

Trinomials: have three terms

**Remember that terms are separated by + and - signs**

Example 1

Classify the polynomial as a monomial, a binomial, or atrinomial: 3xy

Since this polynomial has one term, it is a monomial .

Example 2

Classify the polynomial as a monomial, a binomial, or atrinomial: -2x + 4

Since this polynomial has two terms, it is a binomial.

Example 3

Classify the polynomial as a monomial, a binomial, or atrinomial: x ² + 2x - 4

Since this polynomial has three terms, it is a trinomial .

Example 4

Classify the polynomial as a monomial, a binomial, or atrinomial: - 2 x y ² z ³

Since this polynomial has one term, it is a monomial.

Degree of a Term is the sum of the exponentsof the variables.

The Degree of a Polynomial is the highestdegree of its terms.

Example 5

Identify the degree of each term and the degree of thepolynomial: - 2 x y ² z ³

This polynomial has one term. To find the degree of the term,we add the exponents of the variables. The variables are x, y,and z. The exponents of these variables are 1, 2, and 3. We hadthese together to get 6. 6 is the degree of the term . Sincethere is only one term, the degree of the polynomial will be 6also.

Example 6

Identify the degree of each term and the degree of thepolynomial: 9x ⁶ y ⁵ - 7x ⁴ y³ + 3x ³ y ⁴ + 17x - 4

This polynomial has five terms. To find the degree of eachterm, we add the exponents of the variables. Let's take it oneterm at a time.

The degree of the first term will be 11 (we add the exponentsof the variables 5+6=11)

The degree of the second term will be 7 (we add the exponentsof the variables 4+3=7)

The degree of the third term will be 7 (we add the exponentsof the variables 3+4=7)

The degree of the fourth term will be 1 (the only exponent inthis term is 1)

The degree of the fifth term will be 0 (this term has novariables so its degree is 0)

The degree of the polynomial will be 11 since 11 is thehighest degree of the terms.

Example 7

Identify the degree of each term and the degree of thepolynomial: 8xy + 9x ² y ² + 2x ³y ³

This polynomial has three terms. To find the degree of eachterm, we add the exponents of the variables. Let's take it oneterm at a time.

The degree of the first term will be 2 (we add the exponentsof the variables 1+1=2)

The degree of the second term will be 4 (we add the exponentsof the variables 2+2=4)

The degree of the third term will be 6 (we add the exponentsof the variables 3+3=6)

The degree of the polynomial will be 6 since 6 is the highestdegree of the terms.

To put a polynomial in Descending Order wearrange the terms in order from the highest exponent down to thelowest exponent. We are only concerned with the first variable ifthe polynomial has more than one variable.

Example 8

Put the polynomial in descending order for x: 3x + 2x ²- 4

We need to re-arrange the terms from the highest exponent tothe lowest. 2 is the highest exponent so we put 2x ²first. The next highest exponent is 1 so we put the 3x next. The-4 will go last: 2x ² + 3x - 4

Example 9

Put the polynomial in descending order for x: 6x ² y- 4x ³ y ⁴ - 3x y ²

We need to re-arrange the terms from the highest exponent tothe lowest. We have two variables, but we are only concernedabout the x. 3 is the highest exponent of x so we put - 4x ³y ⁴ first. The next highest exponent of x is 2 so weput the 6x ² y next. The - 3x y ² xy willgo last: - 4x ³ y ⁴ + 6x ² y -3x y ²