Naming Polynomials

8.1Part 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 : monomialHere are some monomials…3x²7xyx8½x

2 terms : BinomialHere are some binomials…5x + 13x² - 4x + y

3 terms : TrinomialHere are some trinomials…7x² + 2x – 10

4 or more terms – polynomialThere 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

Name each expression based on its number of terms

1. 5x + 1 2. 7x²3. 5x – 2xy + 3y4. 6x³ - 9x² + x – 10

1. 5x + 1 Binomial 2. 7x² Monomial3. 5x – 2xy + 3y Trinomial4. 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 DegreeThe degree of a monomial is the sum of the exponents of its variables.

For example,The degree of 7x³ is 3The degree of 8y²z³ is 5The degree of -10xy is 2The 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. 7x2. x² + 3x – 13. 104. 9x²y³5. 12 – 13x³ + 4x + 5x²

1. 7x 12. x² + 3x – 1 23. 10 04. 9x²y³ 55. 12 – 13x³ + 4x + 5x² 3

Names of Polynomials by their Degree

Degree of 0 : ConstantFor example,7

-10

8

Degree of 1 : LinearFor example,

3x – 2½x + 712x – 1

Degree of 2 : QuadraticFor example,7x² - 3x + 64x² - 1

Degree of 3 : Cubic

For example,8x³ + 5x +92x³ - 11

Anything with a degree of 4 or more does not have a special name

Examples

Name each Polynomial by its degree.

1. 10x³ + 2x2. 3x + 83. 64. 9x² + 3x – 15.

1. 10x³ + 2x Cubic2. 3x + 8 Linear3. 6 Constant4. 9x² + 3x – 1 Quadratic5. Not a polynomial!

Putting it all together…ExamplesClassify each polynomial based on its degree and the number of terms:1. 7x³ - 10x 2. 8x – 43. 4x² + 11x – 24. 10x³ + 7x² + 3x – 55. 66. 3x² - 4x

1. 7x³ - 10x cubic/binomial2. 8x – 4 linear/binomial3. 4x² + 11x – 2 quadratic/trinomial4. 10x³ + 7x² + 3x – 5 cubic/polynomial5. 6 constant/monomial6. 3x² - 4x 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 + 3x²2. 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

Copy the table and fill in the blanks.Polynomial Degree Name by

DegreeNumber of Terms

Name by Terms

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