- Polynomials
- Classification of Polynomials
- Classification of Polynomial Equations

Inside Collection (Textbook): Elementary Algebra

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form. Objectives of this module: be familar with polynomials, be able classify polynomials and polynomial equations.

- Polynomials
- Classification of Polynomials
- Classification of Polynomial Equations

Let us consider the collection of all algebraic expressions that do not contain variables in the denominators of fractions and where all exponents on the variable quantities are whole numbers. Expressions in this collection are called polynomials.

Some expressions that *are* polynomials are

A fraction occurs, but no variable appears in the denominator.

Some expressions that *are not* polynomials are

A variable appears in the denominator.

A negative exponent appears on a variable.

Polynomials can be classified using two criteria: the number of terms and degree of the polynomial.

Number of Terms | Name | Example | Comment |

One | Monomial | mono means “one” in Greek. | |

Two | Binomial | bi means “two” in Latin. | |

Three | Trinomial | tri means “three” in Greek. | |

Four or more | Polynomial | poly means “many” in Greek. |

The degree of a term containing only *one* variable is the value of the exponent of the variable. Exponents appearing on numbers do not affect the degree of the term. We consider only the exponent of the variable. For example:

8 is a monomial of degree 0. We say that a nonzero number is a term of 0 degree since it could be written as

The degree of a term containing *more* than one variable is the *sum* of the exponents of the variables, as shown below.

The degree of a polynomial is the degree of the *term* of highest degree; for example:

Polynomials of the first degree are called linear polynomials.

Polynomials of the second degree are called quadratic polynomials.

Polynomials of the third degree are called cubic polynomials.

Polynomials of the fourth degree are called fourth degree polynomials.

Polynomials of the

Nonzero constants are polynomials of the 0th degree.

Some examples of these polynomials follow:

43 is a 0th degree polynomial.

As we know, an equation is composed of two algebraic expressions separated by an equal sign. If the two expressions happen to be polynomial expressions, then we can classify the equation according to its degree. Classification of equations by degree is useful since equations of the same degree have the same type of graph. (We will study graphs of equations in Chapter 6.)

The degree of an equation is the degree of the highest degree expression.

This is a linear equation since it is of degree 1, the degree of the expression on the left of the

Classify the following equations in terms of their degree.

first, or linear

quadratic

cubic

linear

linear

quadratic

quadratic

linear

eighth degree

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

5

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it.

linear

quadratic

linear

cubic

quadratic

linear

linear

cubic

fifth degree

19th degree

The expression

The expression

. . . there is a variable in the denominator

Is every algebraic expression a polynomial expression? If not, give an example of an algebraic expression that is not a polynomial expression.

Is every polynomial expression an algebraic expression? If not, give an example of a polynomial expression that is not an algebraic expression.

yes

How do we find the degree of a term that contains more than one variable?

*((Reference))* Use algebraic notation to write “eleven minus three times a number is five.”

*((Reference))* Simplify

*((Reference))* Find the value of

*((Reference))* List, if any should appear, the common factors in the expression

*((Reference))* State (by writing it) the relationship being expressed by the equation

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Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"