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# Polynomials

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 of monomials.

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 a trinomial: 3xy

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

Example 2

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

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

Example 3

Classify the polynomial as a monomial, a binomial, or a trinomial: 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 a trinomial: - 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 exponents of the variables.

The Degree of a Polynomial is the highest degree of its terms.

Example 5

Identify the degree of each term and the degree of the polynomial: - 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 had these together to get 6. 6 is the degree of the term . Since there is only one term, the degree of the polynomial will be 6 also.

Example 6

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

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

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

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

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

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

The degree of the fifth term will be 0 (this term has no variables so its degree is 0)

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

Example 7

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

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

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

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

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

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

To put a polynomial in Descending Order we arrange the terms in order from the highest exponent down to the lowest exponent. We are only concerned with the first variable if the 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 to the lowest. 2 is the highest exponent so we put 2x 2 first. 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 to the lowest. We have two variables, but we are only concerned about the x. 3 is the highest exponent of x so we put - 4x 3 y 4 first. The next highest exponent of x is 2 so we put the 6x 2 y next. The - 3x y 2 xy will go last: - 4x 3 y 4 + 6x 2 y - 3x y 2